Twisted Resurgence Monomials and canonical-spherical synthesis of Local Objects.

August 21, 2003

Contents

1 Introduction: Object Analysis and Object Synthesis. 4 1.1 The notion of Local Analytic Object. . . . . . . . . . . . . . . 5 1.2 Object Analysis: the Bridge Equation. . . . . . . . . . . . . . 7 1.3 Object Synthesis: semi-formal candidates . . . . . . . . . . . . 9 1.4 Object Synthesis: from semi-formal to effective. . . . . . . . . 10

2 Reminders about moulds, resurgent functions, alien derivations. 11 2.1 Moulds/comoulds. . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Basic mould operations. . . . . . . . . . . . . . . . . . 11 2.1.2 Basic mould symmetries. . . . . . . . . . . . . . . . . . 12 2.1.3 Mould-comould contractions. . . . . . . . . . . . . . . 12 2.1.4 Arborification and co-arborification. . . . . . . . . . . . 13 2.1.5 Armoulds and ormoulds. Operations and symmetries. . 14 2.1.6 Constant-type moulds. . . . . . . . . . . . . . . . . . . 15 2.1.7 Difference-type flat moulds. . . . . . . . . . . . . . . . 16 2.1.8 Difference-type polar moulds. . . . . . . . . . . . . . . 16 2.1.9 Sum-type flat moulds. . . . . . . . . . . . . . . . . . . 17 2.1.10 Sum-type polar moulds. . . . . . . . . . . . . . . . . . 17 2.1.11 Some mould properties for future use. . . . . . . . . . 18 2.1.12 From alternel/symmetrel to alternal/symmetral. . . . . 19

2.2 Resurgent functions. . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Alien derivations or automorphisms. Their weights. . . . . . . 23 2.4 Resurgence monomials. . . . . . . . . . . . . . . . . . . . . . . 25 3 Object Analysis: six basic examples. 27

3.1 Example 1: shift-like diffeomorphism. . . . . . . . . . . . . . . 27 3.2 Example 2: Euler-like differential equation. . . . . . . . . . . . 28 3.3 Example 3: monocritical linear differential system. . . . . . . . 29 3.4 Example 4: monocritical non-linear differential system. . . . . 29 3.5 Example 5: polycritical linear differential system. . . . . . . . 30 3.6 Example 6: polycritical non-linear differential system. . . . . . 31

4 The reverse problem: Object Synthesis. 31 4.1 Standard or hyperlogarithmic resurgence monomials and monics. 32 4.2 Semi-formal synthesis in Example 1. . . . . . . . . . . . . . . 32 4.3 Semi-formal synthesis in Example 2. . . . . . . . . . . . . . . 33 4.4 Semi-formal synthesis in the remaining examples. . . . . . . . 34 4.5 Inadequacy of the standard resurgence monomials for the pur-

pose of synthesis. . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5.1 Unary case: . . . . . . . . . . . . . . . . . . . . . . . . 36 4.5.2 Binary case: . . . . . . . . . . . . . . . . . . . . . . . . 36 4.5.3 Unilateral case: . . . . . . . . . . . . . . . . . . . . . . 36 4.5.4 Sesquilateral case: . . . . . . . . . . . . . . . . . . . . . 37 4.5.5 Bilateral case: . . . . . . . . . . . . . . . . . . . . . . . 37

4.6 First intimations of “antipodality”: . . . . . . . . . . . . . . . 37 4.7 The need for one free parameter at least. . . . . . . . . . . . . 40

5 Methods for non-canonical Object Synthesis. 40 5.1 Main and earliest method. . . . . . . . . . . . . . . . . . . . . 40 5.2 Malgrange’s method. . . . . . . . . . . . . . . . . . . . . . . . 42 5.3 The quasiconformal method. . . . . . . . . . . . . . . . . . . . 43 5.4 Comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Four closely related challenges. 44 6.1 The main obstacle: faster-than-lateral growth. . . . . . . . . . 44 6.2 Challenge 1 : Searching for well-behaved averages. . . . . . . . 46 6.3 Challenge 2 : Searching for well-behaved alien derivations… 50 6.4 Challenge 3 : Searching for well-behaved resurgence monomials. 53 6.5 Challenge 4 : Searching for the proper notion of ramified ex-

ponential growth. . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.6 Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 Well-behaved uniformising averages. 61 7.1 Reminder about the standard or median average. . . . . . . . 61 7.2 Diffusion-induced averages. . . . . . . . . . . . . . . . . . . . . 61 7.3 Scale-invariance. . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7.4 The standard and organic averages as limit-cases. . . . . . . . 63 7.5 Proofs and comments. . . . . . . . . . . . . . . . . . . . . . . 64 7.6 When exactly must we take recourse to well-behaved averages ? 66

8 Well-behaved alien derivations. 69 8.1 Reminder about the standard alien derivations. . . . . . . . . 69 8.2 Diffusion-induced alien derivations. . . . . . . . . . . . . . . . 69 8.3 Scale-invariance. . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.4 Standard and organic alien derivations as limit-cases. . . . . . 70 8.5 Proofs and comments. . . . . . . . . . . . . . . . . . . . . . . 71 8.6 Tables of averages and derivations. . . . . . . . . . . . . . . . 73 8.7 Pinpointing the difference between “good” and “bad”. . . . . . 76

9 Proper notion of ramified-exponential growth. 77 9.1 Bounded singularities. . . . . . . . . . . . . . . . . . . . . . . 77 9.2 Manageable singularities. . . . . . . . . . . . . . . . . . . . . . 78 9.3 Arbitrary singularities. An open question. . . . . . . . . . . . 78

10 Well-behaved resurgence monomials. 79 10.1 Reminder about the standard resurgence monomials. . . . . . 79 10.2 Well-behaved resurgence monomials induced by prodiffusions. 80 10.3 Canonical well-behaved resurgence monomials. . . . . . . . . . 81 10.4 Proofs and comments. . . . . . . . . . . . . . . . . . . . . . . 82 10.5 Extension to the multicritical case. . . . . . . . . . . . . . . . 84

11 Applications to canonical Object Synthesis. 86 11.1 Outline of the construction. . . . . . . . . . . . . . . . . . . . 86 11.2 Proof of the convergence for a positive or large enough twist c. 87 11.3 What is so special about the twistless case ? . . . . . . . . . 92 11.4 The antipodal pairing. . . . . . . . . . . . . . . . . . . . . . . 94 11.5 Iso-invariant deformations. . . . . . . . . . . . . . . . . . . . . 95 11.6 Remarks and complements. . . . . . . . . . . . . . . . . . . . 95

12 Hyper-, peri-, para-logarithmic monomials and monics: total closure. 97 12.1 Main objects and notations. . . . . . . . . . . . . . . . . . . . 97

12.1.1 Hyperlogarithms, perilogarithms, paralogarithms. … 97 12.1.2 ∂- or Δ-friendly monomials and monics. . . . . . . . . 97 12.1.3 Total closure. . . . . . . . . . . . . . . . . . . . . . . . 97

12.2 Hyperlogarithmic monomials and monics. . . . . . . . . . . . . 98 12.2.1 Basic hyperlogarithms. . . . . . . . . . . . . . . . . . . 98 12.2.2 Basic relations. . . . . . . . . . . . . . . . . . . . . . . 98 12.2.3 More relations. . . . . . . . . . . . . . . . . . . . . . . 98 12.2.4 Ordinary and alien differentiation. The ∂ ↔ Δ duality . 99

12.3 Perilogarithmic monomials and monics. . . . . . . . . . . . . . 99 12.3.1 Basic perilogarithms. . . . . . . . . . . . . . . . . . . . 99 12.3.2 Basic relations. . . . . . . . . . . . . . . . . . . . . . . 100 12.3.3 Integral formulae for the Δ-friendly monomials and

monics. . . . . . . . . . . . . . . . . . . . . . . . . . . 100 12.3.4 More relations for the Δ-friendly perilogarithms. . . . . 101 12.3.5 Yet more relations for the Δ-friendly perilogarithms. . 102 12.3.6 A glimpse of the ∂-friendly perilogarithms. . . . . . . . 103 12.3.7 From ∂- to Δ-friendly . . . . . . . . . . . . . . . . . . 103 12.3.8 Resurgence equations. . . . . . . . . . . . . . . . . . . 103

12.4 Paralogarithmic monomials and monics. . . . . . . . . . . . . 104

Abstract. Until very recently, received wisdom seemed to rule out the feasibility of a truly canonical and completely explicit synthesis for local objects, that is to say the possibility of constructing privileged representatives in each analytic conjugacy class of such objects. But in the mid 90s it emerged that there does exist a canonical synthesis after all. We call it paralogarithmic, because its building bricks are a new class of transcendental functions, the so-called paralogarithms, quite distinct from the classical but (in this context) unsuitable hyperlogarithms. We also call it spherical, because the most salient feature of the objects thus produced is a tendency to extend to the whole Riemann sphere (in the critical variable) and to go in pairs: a direct object and its antipodal reflection. Both objects – direct and antipodal – always exist; they are indisputably canonical upto the choice of one unremovable parameter (the “twist” c); and they connect under analytic continuation “whenever the invariants permit”.

1 Introduction: Object Analysis and Object Synthesis.

Although the proper ambit of this paper is canonical object synthesis and the special functions – technically known as as well-behaved resurgence monomials – on which synthesis relies, we have chosen to add a cursory treatment of three closely related topics : well-behaved convolution averages, well-behaved alien derivations, and ramified-exponential growth. Also, to make the whole thing tolerably self-contained, preliminary sections have been inserted, which recall the basic notions about moulds and resurgence. This apparatus is light, easy to master, and altogether a good investment. It makes possible a far-going algebraisation of the analysis difficulties. It also permits explicit constructions, and leads to proofs that are often quite concise.

The results about canonical synthesis go back to March 1996 (they were presented in a general survey I gave at the Orsay conference in honour of A. Douady in June 1996) but nothing in the way of a systematic written exposition has appeared so far ^{1}. This, I feel, is one more reason for attempting a fairly comprehensive treatment this time around.

1.1 The notion of Local Analytic Object.

By local analytic object we shall mean, primarily :

(1) germs of singular analytic vector fields at 0 on ℂ^{ν}, often referred to as just fields for short

(2) germs of analytic diffeomorphisms of ℂ^{ν} into itself, with 0 as fixed point, or diffeos for short and, secondarily, all those equations or systems (differential, difference, functional, etc) which may, in a standard manner, be rephrased in terms of fields or diffeos.

Fields will be noted

X = ∑_{1≤i≤ν} X_{i}(x) ∂_{x_{i}}  ;  X_{i}(x) ∈ ℂ{x_{1}, …, x_{ν}}  (1)

but instead of diffeos proper:

f : x_{i} ↦ f_{i}(x)  i = 1, …, ν ; f_{i}(x) ∈ ℂ{x_{1}, …, x_{ν}} (2)

it will often be more convenient to handle the corresponding substitution operators F (same symbols, but capitalised):

(F φ)(x) := φ(f(x)) ;  ∀φ(x) ∈ ℂ{x_{1}, …, x_{ν}}  (3)

The discussion hinges on the nature of the object’s spectrum, ie the eigenvalues of its linear part: λ = (λ_{1}, …, λ_{ν}) for a field; l = (l_{1}, …, l_{ν}) for a diffeo. If the spectrum is “generic”, then the object is analytically conjugate to its linear part – and that ends the matter, at least from the local point of view.

1though some sketchy indications found their way into [EM], 1996.

Difficulties arise only in the case of

– resonance : relations of type 0 = λ_{i} − ∑ λ_{j} or 0 = l_{i} − ∏(l_{j})^{m_{j}} with non-negative integers m_{j}.

– quasiresonance : when Bryuno’s well-known diophantine condition, which minorises the above expressions in terms of ∥m∥, is not fulfilled.

– nihilence : this condition, which presupposes resonance but bears on coefficients of all orders, occurs mostly in connection with symplectic or volumepreserving objects.

The more “complicated” an object, the larger its set of invariants tends to be. Alongside the formal and analytic invariants (ie relative to formal or analytic coordinate changes) we have the notion of holomorphic invariants – ie those invariants that depend holomorphically on the object Ob (ie in practical terms, on its Taylor coefficients), at least when Ob remains within a fixed formal conjugacy class. ^{2}

Resonance generates formal invariants (other that the spectrum itself), of which there may be an infinite number^{3} if the resonance degree is ≥ 2.

Each of the above-mentioned complications – resonance, quasiresonance, nihilence – whether in isolation or in combination, gives rise to analytic invariants (strictly analytic, ie non formal). Moreover, when resonance alone is at work, there tend to exist ^{4} complete systems of analytic-cum-holomorphic invariants {𝔸_{ω}}.

We shall henceforth deal with purely resonant objects. Resonance is more important, and less exceptional, than “Lebesgue measure” considerations would suggest, since it covers the cases :

(1) of identity-tangent diffeomorphisms

(2) of vector fields with (one or several) vanishing eigenvalues and also since

(3) most singular differential equations or systems, when translated into timeautonomous systems (by the addition of one unknown) or, equivalently, into vector fields, tend to exhibit resonance.

“Object analysis” starts from some object Ob and is concerned with finding its invariants. For resonant objects, which alone matter to us here, there is a method of sweeping generality – the Bridge equation (see infra) – for conctructing complete systems {𝔸_{ω}} of analytic-cum-holomorphic invariants, in the form of specific differential operators 𝔸_{ω}, with indices ω running through a countable set Ω generated by the object’s spectrum. Moreover, relatively to any “suitable” basis {Δ Δ_{ω}^{good}} in the algebra of alien derivations, the Bridge equation yields systems {𝔸_{ω}^{good}} that can be characterised by means of simple, transparent growth conditions on the invariants 𝔸_{ω}^{good} as ω increases.

2thus, when applied to invariants, the words analytic and holomorphic assume quite distinct meanings: analytic invariants are not necessarily holomorphic.

3more precisely, an infinite number of independent formal invariants

4with slight qualifications, see eg [E5]

“Object synthesis” is the converse problem: starting from a prescribed system {𝔸_{ω}} with the proper growth pattern, find an object Ob whose invariants coincide with that system. There are actually four degrees there :

– existence : showing that such an object Ob does exist.^{5}

– constructiveness : producing an effective procedure for constructing it.

– expliciteness : expanding the object Ob, in a manner both explicit and universal, by means of elementary special functions, the so-called resurgence monomials, that are not constructed ad hoc, but are given once and for all.

– canonicity : examining whether perchance there exists a “canonical” choice for Ob and also (since we don’t want to forego expliciteness) a corresponding system of “canonical” resurgence monomials.

We shall recall (§5) the basic facts about existential, constructive, explicit synthesis, and sketch several methods, some going back to the late 70s, for achieving these. But our real concern here is with the more ambitious goal of explicit-canonical synthesis. We shall recall earlier attempts in this direction, based on the notion of hyperlogarithmic monomials, and show why these attempts, while interesting in their way and insightful, were doomed to partial failure. We shall then (§8) introduce a whole new class of resurgence monomials, based on “prodiffusions”, and which on account of their nice growth properties, do permit explicit synthesis in all cases. Lastly, we shall (§§8,9,10) show that there exists a particular subclass, the so-called paralogarithmic or spherical monomials, which unquestionably stand out as “canonical” and which can be harnessed to synthesise objects Ob that will automatically inherit their ‘canonicity’.

1.2 Object Analysis: the Bridge Equation.

Let Ob be some (purely) resonant object – field or diffeo – expressed in a particular analytic chart x = {x_{1}, …, x_{ν}} that diagonalises the object’s linear part. The object’s complete linearisation is usually impossible, even formally, and what takes its place is either formal normalisation, which removes all but a few resonant monomials, or the more radical step of formal trivialisation, which forfeits entireness ^{6} but reduces the object to the simplest conceivable form, namely ∂_{z} for a field and z ↦ z + 1 for a diffeo.

5or, what amounts to the same, finding conditions for a system {𝔸_{ω}} to be “admissible”, ie to be someone’s invariants.

Let y = {y_{1}, …, y_{ν}} be a formal normal chart, and consider the formalentire coordinate changes y_{i} = θ_{i}(x) and x_{i} = θ_{i}^{−1}(y) with the substitution operators Θ and Θ^{−1} that go with them : Θ^{±}φ := φ ∘ θ^{±}

Consider also the “trivial chart” {z, u} = {z, u_{1}, …, u_{ν−1}}. Expressing the given coordinates x = {x_{i}} in terms of the trivial coordinates {z, u_{i}}, we get the so-called formal integral x(z, u) = {x_{1}(z, u), …, x_{ν}(z, u)}, which verifies:

∂_{z} x(z, u) ≡ X(x(z, u))  for a field  (4)
f(x(z, u)) ≡ x(z + 1, u)  for a diffeo  (5)

The Bridge Equation (B.E.) is an amazingly general and flexible tool for extracting the object’s invariants from the divergence-resurgence of the trivialising or (direct/inverse) normalising transformations. Here are its three main forms :

Δ Δ_{ω} x(z, u) = 𝔸_{ω} x(z, u)  (B.E. for formal integral)  (6)
[Δ Δ_{ω}, Θ]  = − Θ 𝔸_{ω}  (B.E. for direct normaliser)  (7)
[Δ Δ_{ω}, Θ^{−1}] = + 𝔸_{ω} Θ^{−1}  (B.E. for inverse normaliser) (8)

The indices ω on both sides of the Bridge Equation range through a countable set Ω essentially spanned by the object’s multipliers, ie the λ_{j} in the case of a field, and the log(l_{j}) (to which one must add the universal multiplier λ_{0} := 2πi ) in the case of a diffeo.

The Δ Δ_{ω} on the left-hand side denotes the alien derivation relative to the variable z and the index ω but with a built-in exponential factor Δ Δ_{ω} := e^{−ω z}Δ_{ω} that makes it commute with ∂_{z} and ensures the stability rule under analytic changes of equivalent variables z ↦ z_{⋆} with z ∼ z_{⋆} :

φ(z) ≡ φ_{∗}(z_{∗})  ⟹  Δ Δ_{ω}^{(z)} φ(z) ≡ Δ Δ_{ω}^{(z_{∗})} φ_{∗}(z_{∗})  (9)

The alien-differentiation variable z, also known as critical variable^{7}, is always ∼ ∞. In (6) it is simply the z inside the formal integral. In (7) (resp (8)) it is the inverse of some resonant monomial, ie z := 1/x^{m} (resp z := 1/y^{m}). Due to the afore-mentioned invariance property of alien differentiation, the critical variable is actually defined up to equivalence ∼ and so the proper, intrinsic notion is that of critical class.

6it no longer relies on entire power series

7or critical time.

The 𝔸_{ω} on the right-hand side are ordinary (first-order) differential operators – in the variables (z, u_{i}) or (x_{i}) or (y_{i}) respectively. They are constructively determined, even overdetermined, by the requirement of equality in the Bridge Equation – whichever of its variants we choose to work with, and whichever critical variable we pick (within the critical class) for alien differentiation. Each single 𝔸_{ω} is an invariant of the object Ob, and the total collection {𝔸_{ω}, ω ∈ Ω} constitutes a set, both complete and free, of analytic-cum-holomorphic invariants.

All these claims, as sketchy as they are sweeping, clearly cry for explanations, and qualifications too, which cannot be supplied here but are available in the literature ([E2],[E3],[E7]). We recalled these statements simply to serve as a general backdrop for the twin problems of object analysis and synthesis – but we shall illustrate the method on just six typical instances.

1.3 Object Synthesis: semi-formal candidates

Pick a purely resonant and, for simplicity, monocritical^{8} object Ob with invariants {𝔸_{ω}} and consider the expansions:

Θ  :=^{always} 1 + ∑_{1≤r} ∑_{ω_{i}∈Ω}(−1)^{r} Ue^{ω_{1},...,ω_{r}} 𝔸_{ω_{r}} … 𝔸_{ω_{1}}  (10)
Θ^{−1} :=^{conditionally} 1 + ∑_{1≤r} ∑_{ω_{i}∈Ω}  Ue^{ω_{r},...,ω_{1}} 𝔸_{ω_{r}} … 𝔸_{ω_{1}}  (11)

Formally, ie ignoring all questions of convergence, they verify the Bridge equation: if we alien-differentiate them term by term, we can see that the operator Θ verifies (7) and its inverse Θ^{−1} inverse verifies (8)^{9}. But (7) and (8) do characterise Θ and Θ^{−1} as direct and inverse normalisers. Thus, working our way back from the normal form Ob^{norm}, we get a new object Ob := Θ Ob^{nor} Θ^{−1} which not only has the prescribed invariants 𝔸_{ω} but is also ‘formally analytic’, in the sense that all its alien derivatives in z – by assumption the only critical time – identically vanish.

8ie such that the associated objects – normalising transformations, fractional iterates, etc – have only one critical time z, or rather critical time class {z}.

9That much is always true. The cautionary word conditionally in the middle of (11)

signals that this second expansion, unlike the first, holds only when the invariants 𝔸_{ω} commute with ∂_{z} , which is always the case with vector fields. But even when there is no commutation, Θ^{−1} admits an analoguous expansion – and knowing Θ is sufficient anyway.

1.4 Object Synthesis: from semi-formal to effective.

But what is missing in this formal, or should we say semi-formal, construction, is the convergence of Θ^{±1} is the relevant space, ie the space of resurgent functions with exponential growth in the Borel plane. Such convergence, which would automatically ensure the convergence of the synthesised object in its own space, namely that of multiplicative function-germs , has itself three sides to it:

(i) convergence on each compact subset of the ramified Borel-plane, ie on the Riemann function generated by singularities from Ω which puncture that plane. (ii) proper growth at infinity, ie exponential growth away from, or parallel to, the singularity-carrying axes, and “suitable” growth (technically :“ramifiedexponential” growth, see §9 ), over such axes. (iii) for the synthesis construction to succeed, all this should apply, if not to the expansions (10), (11) themselves, at least to some re-arranged variant of these expansions.

Demand (i) calls for the introduction not only of well-behaved resurgence monomials, but also of a suitable twist parameter c.

Demand (ii) is then automatically fulfilled: it follows from the growth conditions which the invariants 𝔸_{ω} verify as soon as they are expressed relative to a well-behaved basis of ALIEN.

That leaves (iii): it will turn out that the absolute convergence of (10) or (11) is too strong a demand, except in very simple – linear or affine – situations. In truly general cases, we must not only work with a well-behaved system of resurgence monomials (preferably the canonical system, since we are fortunate in having one) and adjust the twist parameter c, but we must also perform two further adaptations :

– we must work relative to a well-behaved basis of ALIEN (which means expressing, simultaneously, our alien derivations, holomorphic invariants, and resurgence monomials coherently in this basis)

– then we must arborify our expansions (10),(11), ie switch from an indexation by totally ordered sequences to one that relies on a partial order, of arborescent type.

2 Reminders about moulds, resurgent functions, alien derivations.

2.1 Moulds/comoulds.

Moulds are functions of a variable number of variables : they depend on sequences ω := (ω_{1}, …, ω_{r}) of arbitrary length r = r(ω). The sum ∥ω∥ of a sequence is simply ∑_{1}^{r} ω_{i}. Sequences are systematically written in boldface, with upper indexation when such is called for, and with the product denoting concatenation: e.g. ω = ω^{1}. ω^{2}. The elements ω_{i} which make up these sequences are written in normal print, with lower indexation. The sequences themselves are affixed to the moulds as upper indices A^{•} = {A^{ω}}, since moulds are meant to be contracted

A^{•}, B_{•} ↦ < A^{•}, B_{•} > := ∑ A^{ω} B_{ω}

with dual objects (often differential operators or elements of an associative algebra), the so-called comoulds B_{•} = {B_{ω}}, which carry their own indices in lower position.

2.1.1 Basic mould operations.

Moulds may be added, multiplied, composed. Mould addition is what you expect : components of equal length get added. Mould multiplication (mu or ×) is associative, but non-commutative :

C^{•} = A^{•} × B^{•} ⟺ C^{ω} = ∑_{ω = ω^{1}. ω^{2}}A^{ω^{1}}B^{ω^{2}}  (12)

(This includes the trivial decompositions ω = ω.∅ and ω = ∅.ω).

Mould composition (∘) too is associative and non-commutative :

C^{•} = (A^{•} ∘ B^{•}) ⟺ C^{ω} = ∑_{ω = ω^{1} …ω^{s}}A^{∥ω^{1}∥,...,∥ω^{s}∥} B^{ω^{1}} … B^{ω^{s}} (13)

with a sum extending to all possible decompositions of ω into s ≤ r(ω) nonempty factor sequences ω^{i}

The operations (+, ×, ∘) on moulds interact in exactly the same way as their namesakes for power series. Thus (A^{•} × B^{•}) ∘ C^{•} ≡ (A^{•} ∘ C^{•}) × (B^{•} ∘ C^{•}).

2.1.2 Basic mould symmetries.

Nearly all useful moulds fall into a few basic symmetry types. A mould A^{•} is said to be symmetral (resp. alternal) iff :

∑_{ω∈sha(ω^{1},ω^{2})} A^{ω} = A^{ω^{1}}A^{ω^{2}} (resp. 0) ∀ ω^{1} ≠ ∅ , ∀ ω^{2} ≠ ∅ (14)

A mould A^{•} is said to be symmetrel (resp. alternel) iff :

∑_{ω∈she(ω^{1},ω^{2})} A^{ω} = A^{ω^{1}}A^{ω^{2}} (resp. 0) ∀ ω^{1} ≠ ∅ , ∀ ω^{2} ≠ ∅ (15)

Here sha(ω^{1}, ω^{2}) (resp. she(ω^{1}, ω^{2} )) denotes the set of all sequences ω obtained from ω^{1} and ω^{2} under ordinary (resp. contracting) shuffling. In a contracting shuffle, two adjacent indices ω_{i} and ω_{j} stemming from ω^{1} and ω^{2} respectively may coalesce to ω_{ij} := ω_{i}+ω_{j}.

2.1.3 Mould-comould contractions.

Let B_{ω} be the homogeneous components of some local-analytic, ν-dimensional vector field X (resp of the postcomposition operator F associated to some local-analytic ν-dimensional diffeomorphism f) and let

B_{ω} = B_{ω_{1},...,ω_{r}} := B_{ω_{r}} … B_{ω_{1}}  (reversal!)  (16)

The comould B_{•} so defined is said to be co-symmetral (resp co-symmetrel) if its action on a product φ_{1}φ_{2} obeys the Leibniz rule :

B_{ω} (φ_{1} φ_{2}) = ∑ (B_{ω^{1}} φ_{1}) (B_{ω^{2}} φ_{2})  (17)

with a sum extending to all pairs (ω^{1}, ω^{2}) such that ω ∈ sha(ω^{1}, ω^{2}) (resp ω ∈ she(ω^{1}, ω^{2})).

The four main symmetry types admit a simple characterisation in terms of mould-comould contractions :

A^{•} : B_{•} ↦ C_{•} with C_{ω_{0}} := ∑_{∥ω∥=ω_{0}} A^{ω} B_{ω}  (18)

Indeed :

\begin{aligned}A^{•} & : B_{•} & → & C_{•} \\ alternal & : field & → & field \\ symmetral & : field & → & diffeo \\ alternel & : diffeo & → & field \\ symmetrel & : diffeo & → & diffeo \end{aligned}

Most stability properties follow from this interpretation. Thus :

\begin{aligned}symmetral^{•} & × & symmetral^{•} & = & symmetral^{•} \\ symmetrel^{•} & × & symmetrel^{•} & = & symmetrel^{•} \\ alternal^{•} & ∘ & alternal^{•} & = & alternal^{•} \\ symmetrel^{•} & ∘ & symmetrel^{•} & = & symmetrel^{•} \end{aligned}

2.1.4 Arborification and co-arborification.

Straightforward mould expansions ∑_{ω} A^{ω} B_{ω}, which typically pair a symmetral or alternal mould A^{•} with a cosymmetral comould B_{•} ( or a symmetrel or alternel mould A^{•} with a cosymmetrel comould B_{•}) often fail to converge absolutely, ie we often have ∑_{ω} ∥A^{ω} B_{ω}∥ = +∞, although the underlying power series may well be convergent. Fortunately, there is an extremely general method for restoring convergence. In essence, it replaces expansions indexed by totally ordered sequences ω by others whose indices are arborescent sequences ω^{≺} or ω^{≺ ≺} , like this :

∑_{ω}A^{ω} B_{ω} ↦ ∑_{ω^{≺}} A^{ω^{≺}} B_{ω}^{≺}  (ordinary arborification) (19)
∑_{ω}A^{ω} B_{ω} ↦ ∑_{ω^{≺ ≺}} A^{ω^{≺ ≺}} B_{ω}^{≺ ≺}  (contracting arborification) (20)

The dual arborification/coarborification transforms verify :

arborification ⟹

ordinary :  A^{ω^{≺}} := ∑_{ω^{≺}<ω} A^{ω}  (complete definition)
contracting :  A^{ω^{≺ ≺}} := ∑_{ω^{≺ ≺} <<ω} A^{ω}  (complete definition)

coarborification ⟹

ordinary :  B_{ω} := ∑_{ω^{≺}<ω} B_{ω}^{≺}  (mere constraint)
contracting :  B_{ω} := ∑_{ω^{≺ ≺} <<ω} B_{ω}^{≺ ≺}  (mere constraint)

and are devised in such a way as to :

(1) leave the expansions formally unchanged : they amount to a simple redistribution of terms.

(2) drastically reduce the size of the comould part : comoulds typically get divided by a factor of order r! := r(ω)!

(3) prevent a concomitant increase of the mould part : moulds typically retain the same order of magnitude, despite being changed into a sum of almost r! similar terms !

But whereas the reduction (2) is automatic and universal, the non-increase

(3) relies on specific identities, of an algebraic or combinatorial nature, which can never be taken for granted, and yet tend to take place, with providential regularity, whenever we require them !

The ordinary (resp contracting) arborification rule boils down to summing all the terms A^{ω} with totally ordered sequences ω whose order is compatible with the arborescent order of ω^{≺} ( resp with that of ω^{≺ ≺} , but allowing contractions of consecutive elements ω_{i}). The following example should make this amply clear. Assume :

ω_{4}
↗
ω^{≺} ( or ω^{≺ ≺} ) := ω_{1} → ω_{2} → ω_{3}

Then the arborification rules means :

A^{ω^{≺}} := A^{ω_{1}, ω_{2}, ω_{3}, ω_{4}} + A^{ω_{1}, ω_{2}, ω_{4}, ω_{3}} + A^{ω_{1}, ω_{4}, ω_{2}, ω_{3}}
A^{ω^{≺ ≺}} := A^{ω_{1}, ω_{2}, ω_{3}, ω_{4}} + A^{ω_{1}, ω_{2}, ω_{4}, ω_{3}} + A^{ω_{1}, ω_{4}, ω_{2}, ω_{3}} + A^{ω_{1}, ω_{2}, ω_{3}+ω_{4}} + A^{ω_{1}, ω_{2}+ω_{4},ω_{3}}
:= A^{ω^{≺}} + A^{ω_{1}, ω_{2}, ω_{3}+ω_{4}} + A^{ω_{1}, ω_{2}+ω_{4},ω_{3}}

Unlike the arborification identities, the coaborification constraints leave a lot of latitude. However, for any given choice of variables, there exists a ‘canonical’ choice, the so-called ‘homogeneous’ coarborification, which shall be described, and made use of, in §11.2.

2.1.5 Armoulds and ormoulds. Operations and symmetries.

Depending on the nature of the order on the indexing sequence, we have three types of objects :

\begin{aligned}total order & ω & : & moulds & A^{•} \\ arborescent order & ω^{≺} & : & armoulds & A^{•^{≺}} \\ arbitrary partial order & ω^{♯} & : & ormoulds & A^{•^{♯}} \end{aligned}

An arborescent order is one under which each ω_{i} has either one antecedent ω_{i_{−}} or none at all, in which case ω_{i} is declared a root. Under the arborification transform (§2.1.4), moulds induce armoulds. Similarly, moulds (as well as armoulds) induce ormoulds. It turns out that all basic mould operations (unary or binary) and symmetries uniquely and unambiguously extend to both armoulds and ormoulds : that extension is natural not only in the sense of commuting with the induction stronger order → weaker order , but also of preserving all the main properties of the operations or symmetries in question (such as, say, the associativity of × or ∘, or the simple rule for calculating the multiplicative inverse of a symmetral/el mould, namely multiplication by (−1)^{r} and order reversal without/with possible contraction of order-adjacent indices ω_{i}) even when applied to ‘primitive’ armoulds or ormoulds, ie to ar/ormoulds that are not induced by any mould. Although most ar/ormoulds encountered in practice are of the induced sort, and therefore carry no more information than the underlying moulds, the change of perspective nonetheless often proves very convenient.

We shall now proceed to list a number of rather elementary yet very useful moulds and armoulds which are going to occur and recur throughout this paper, and to mention their main properties. These properties are of a combinatorial or algebraic nature but, as usual in our ‘formalisation approach’, they ‘absorb’ and resolve most of the ‘analysis’ or ‘divergence’ difficulties which stand in the way – here, in the way of Object Synthesis.

Let us first settle some nomenclature :

– constant-type moulds depend only on the length r of the sequence ω^{10} – sum-type ( resp difference-type) moulds depend on partial sums (resp differences) of their indices ω_{i}

– flat (resp polar) moulds are expressible as simple superpositions of the sign function (resp the polar function t ↦ 1/t).

2.1.6 Constant-type moulds.

mould value  symmetry type associated series
1^{•}  1 if r = 0 (0 otherwise) symmetral  1
I^{•}  1 if r = 1 (0 otherwise) alternal  x
log^{•}  \frac{(−1)^{r−1}}{r}  alternel  log(1 + x)
exp_{a}^{•}  \frac{a^{r}}{r!}  symmetral  e^{a x}
tu_{a}^{•}  \frac{(−1)^{r}}{r!} \frac{Γ(r−a)}{Γ(−a)}  symmetrel  (1 + x)^{a}

10they correspond one-to-one to power series M^{•} ↦ M^{∅} + ∑_{1≥r} M\overbrace{1, …, 1}^{r times} x^{r} and this correspondance commutes with all three operations +, ×, ∘.

2.1.7 Difference-type flat moulds.

sad^{∅}  := 1
sad^{t_{1},...,t_{r}} := 1 if t_{1} < t_{2} < ⋯ < t_{r}
sad^{t_{1},...,t_{r}} := 0 otherwise
lad^{∅}  := 0
lad^{t_{1},...,t_{r}} := (−1)^{q} \frac{p! q!}{(p + q + 1)!} = (−1)^{q} \frac{p! q!}{r!}
with  p := ∑_{t_{i}<t_{i+1}} 1 and q := ∑_{t_{i}>t_{i+1}} 1

2.1.8 Difference-type polar moulds.

tas_{a,b}^{∅}  := 1
tas_{a,b}^{t_{1}}  := \frac{a − b}{(a − t_{1})(t_{1} − b)}
tas_{a,b}^{t_{1},...,t_{r}} := \frac{a − b}{(a − t_{1})(t_{1} − t_{2}) … (t_{r−1} − t_{r})(t_{r} − b)}

tas_{⋆}^{∅}  := 0
tas_{⋆}^{t_{1}}  := \frac{1}{(−t_{1})(t_{1})}
tas_{⋆}^{t_{1},...,t_{r}} := \frac{1}{(−t_{1})(t_{1} − t_{2}) … (t_{r−1} − t_{r})(t_{r})}

tas_{⋆⋆}^{∅}  := 0
tas_{⋆⋆}^{t_{1}}  :=  1
tas_{⋆⋆}^{t_{1},...,t_{r}} := \frac{1}{(t_{1} − t_{2}) … (t_{r−1} − t_{r})}

tas_{a,b}^{•} × tas_{b,c}^{•} = tas_{a,c}^{•}
tas_{a,b}^{•} × tas_{b,a}^{•} = 1^{•}

2.1.9 Sum-type flat moulds. Some abbreviations first :

x := (x_{1}, …, x_{r})  (21)
x̌_{i} := x_{1} + ⋯ + x_{i}  (22)
x̂_{i} := x_{i} + ⋯ + x_{r}  (23)
∥x∥ := x_{1} + ⋯ + x_{r} = x̂_{1} = x̌_{r}  (24)
σ_{+}(x) := 1 if x > 0 (resp := 0 if x < 0)  (25)
σ_{−}(x) := 1 if x < 0 (resp := 0 if x > 0)  (26)
δ(x) := 1 if x = 0 (resp := 0 if x ≠ 0)  (27)
sofo_{±}^{x} :=  (−1)^{r} σ_{±}(x̌_{1}) … σ_{±}(x̌_{r})
antisofo_{a}^{x} :=  (−1)^{r} σ_{±}(x̂_{1}) … σ_{±}(x̂_{r})
sefo_{±}^{x} := (−1)^{r−1}σ_{±}(x̌_{1}) … σ_{±}(x̌_{r−1}) σ_{∓}(x̌_{r})
antisefo_{±}^{x} := (−1)^{r−1}σ_{∓}(x̂_{1}) σ_{±}(x̂_{r−1}) … σ_{±}(x̂_{r})
lefo_{±}^{x} :=  (−1)^{r}σ_{±}(x̌_{1}) … σ_{±}(x̌_{r−1}) δ(x̌_{r})
antilefo_{±}^{x} :=  (−1)^{r}δ(x̂_{1}) σ_{±}(x̂_{r−1}) … σ_{±}(x̂_{r})

2.1.10 Sum-type polar moulds.

sa_{a}^{ω}  := ∏_{i=1}^{i=r} \frac{ω_{i}}{ω̌_{i}}  musa_{a}^{ω}  := (−1)^{r} ∏_{i=1}^{i=r} \frac{ω_{i}}{ω̂_{i}}
romo_{a}^{ω} := ∏_{i=1}^{i=r}(a \frac{ω_{i}}{ω̌_{i}} − 1)  antiromo_{a}^{ω} := ∏_{i=1}^{i=r}(a \frac{ω_{i}}{ω̂_{i}} − 1)
remo_{a}^{ω} := a \frac{ω_{r}}{ω̌_{r}} ∏_{i=1}^{i=r−1}(a \frac{ω_{i}}{ω̌_{i}} − 1) antiremo_{a}^{ω} := a \frac{ω_{1}}{ω̂_{1}} ∏_{i=2}^{i=r}(a \frac{ω_{i}}{ω̂_{i}} − 1)
somo_{a,b}^{•} := remo_{a}^{•} × antiromo_{1−b}^{•}  (28)
:= romo_{a}^{•} × antiremo_{1−b}^{•}  (29)
= romo_{a/b}^{•} × remo_{b}^{•}  (30)
somo_{[ {a \atop c} {b \atop d}]}^{•} := somo_{\frac{c−b}{d−b} , \frac{a−b}{d−b}}^{•}  (31)
somo_{[ {b \atop a} {0 \atop 1} ]}^{•} := somo_{a,b}^{•}  (32)

2.1.11 Some mould properties for future use. Symmetry types.^{11} All the above moulds fall into one or the other of the main symmetry types.

Alternal : lad^{•}, tas_{⋆}^{•}, tas_{⋆⋆}^{•} Symmetral : exp_{a}^{•}, sad^{•}, tas_{a,b}^{•}, sa^{•}, musa^{•} Alternel : log^{•}, lefo_{±}^{•}, redo_{±}^{•}, redom^{•} Symmetrel : tu_{a}^{•}, sofo_{±}^{•}, sefo_{±}^{•}, romo_{a}^{•}, remo_{a}^{•}, somo_{a,b}^{•} Pairs of the form mould^{•}, antimould^{•} have the same symmetry type.

Useful identities and closure properties :

sofo•_{+} × sefo•_{−} = 1^{•}  (33)
antisofo•_{+} × antisefo•_{−} = 1^{•}  (34)

remo_{a}^{•} × antiromo_{1−a}^{•} = 1^{•}  (35)
romo_{a}^{•} × antiremo_{1−a}^{•} = 1^{•}  (36)
multplicative inverse : somo_{[ {a \atop c} {b \atop d}]}^{•} ↔ somo_{[ {c \atop a} {b \atop d}]}^{•}  ( a , c exchanged )
composition inverse :  somo_{[ {a \atop c} {b \atop d}]}^{•} ↔ somo_{[ {b \atop d} {a \atop c}]}^{•}  ( columns exchanged )
sequence reversal :  somo_{[ {a \atop c} {b \atop d}]}^{•} ↔^{anti} somo_{[ {c \atop a} {d \atop b}]}^{•}  ( rows exchanged )
multplication : somo_{a_{1},a_{2}}^{•} × somo_{a_{2},a_{3}}^{•} = somo_{a_{1},a_{3}}^{•}
composition :  somo_{a_{1},b_{1}}^{•} ∘ somo_{a_{2},b_{2}}^{•}  = somo_{(a_{2}−b_{2})a_{1}+b_{2} , (a_{2}−b_{2})b_{1}+b_{2}}^{•}
multplication : somo_{[ {a_{0} \atop a_{2}} {b_{1} \atop b_{2}} ]}^{•} × somo_{[ {a_{1} \atop a_{0}} {b_{1} \atop b_{2}} ]}^{•} = somo_{[ {a_{1} \atop a_{2}} {b_{1} \atop b_{2}} ]}^{•}
composition :  somo_{[ {a_{1} \atop a_{2}} {b_{1} \atop b_{2}} ]}^{•} ∘ somo_{[ {b_{1} \atop b_{2}} {c_{1} \atop c_{2}} ]}^{•} = somo_{[ {a_{1} \atop a_{2}} {c_{1} \atop c_{2}} ]}^{•}

11flat moulds should be regarded as distribution-valued : for them the symmetries hold almost everywhere, not necessarily everywhere.

Smooth or size-preserving arborification.

All the above moulds possess the property of smooth arborification (meaning that their arborified variants admit essentially the same type of bounds) the only exception being the moulds log^{•} and tu_{a}^{•} for a ∉ ℤ and in particular for a = 1/2. This is in relation with the fact that the standard alien derivations (which admit log^{•} as their left-lateral mould) and the standard or median convolution average (which admits tu_{1/2}^{•} as its right- and left-lateral mould) are not well-behaved.

Of course, for alternal or symmetral (resp alternel or symmetrel) moulds, one should take the ordinary (resp contracting) form of arborification.

Form-preserving arborification.

All the sum-type moulds listed above, ie all those moulds whose definition involves forward sums x̂_{i} or ω̂_{i} (resp backward sums x̌_{i} or ω̌_{i}) have the stronger and very useful property of form-preserving arborification. This means that they retain their outward analytical expression, except that the sums x̂_{i} or ω̂_{i} (resp x̌_{i} or ω̌_{i}) are now relative to the arborescent (resp antiarborescent) order. The same holds for the difference-type moulds tas_{a,∞}^{•} and tas_{∞,b}^{•}.

2.1.12 From alternel/symmetrel to alternal/symmetral.

Let us also mention two mould transforms which turn alternel (resp symmetrel) moulds A^{•} into alternal (resp symmetral) moulds B^{•}. The first transform is quite elementary and applies to all cases. The second transform is more subtle, but also more relevant to the present investigation. It applies only to moulds A^{•} with indices n_{i} in ℕ and turns them into ‘flat’ or ‘piecewise-constant’ moulds B^{•} with indices t_{i} in ℝ. Both transforms respect multiplication in the sense that transf(A_{1}^{•}×A_{2}^{•}) ≡ transf(A_{1}^{•})×transf(A_{2}^{•}). Here is how they are defined :

First mould transform:

direct :  A^{•} ↦ B^{•} := A^{•} ∘ exp_{1}^{•}  (37)
inverse :  B^{•} ↦ A^{•} := B^{•} ∘ log^{•}  (38)

Second mould transform:

A^{•} ↔ B^{•} with B^{t_{1},...,t_{r}} := SA^{ϵ_{1},...,ϵ_{r−1},+} and
ϵ_{1} := sign(t_{1} − t_{2}) , …, ϵ_{r−1} := sign(t_{r−1} − t_{r})
SA^{+} := −A^{1}
SA^{+,+} := +A^{1,1}
SA^{−,+} := +A^{1,1} + A^{2}
SA^{+,+,+} := −A^{1,1,1}
SA^{+,−,+} := −A^{1,1,1} − A^{1,2}
SA^{−,+,+} := −A^{1,1,1} − A^{2,1}
SA^{−,−,+} := −A^{1,1,1} − A^{1,2} − A^{2,1} − A^{3}
etc …
Or generally :
direct :  SA^{ϵ^{1},...,ϵ^{s}} := (−1)^{r} ∑^{⋆} A^{n^{1},...,n^{s}}
inverse :  A^{r_{1},...,r_{s}} := (−1)^{s} ∑^{⋆⋆} ϵ_{1} … ϵ_{r} SA^{ϵ_{1},...,ϵ_{r}}

In the last but one identity, all sign subsequences ϵ^{i} consist of (r_{i} − 1) initial − signs and one final + sign (r_{i} may be 1) and ∑^{⋆} extends to all integer sequences n^{i} of sum r_{i} , whereas in the last (reverse) identity the sum ∑^{⋆⋆} extends to all ϵ_{j} ∈ {+, −} except when j ∈ {r_{1}, r_{1} + r_{2}, …, r_{1} + ⋯ + r_{s}}, in which case ϵ_{j} has to be +.

2.2 Resurgent functions.

Roughly, the algebra of real resurgent fonctions ^{12} consists of all (analytic or cohesive) function germs at +0 that possess an endless (analytic or cohesive, but usually ramified) forward continuation over the whole of ℝ^{+}.

Similarly, the general algebra of complex resurgent fonctions consists of all analytic function germs at 0_{•} that possess an endless (analytic but usually ramified) continuation over the whole of ℂ_{•}. Here, ℂ_{•} denotes the Riemann surface of the logarithm function with its ‘ramified’ origin 0_{•} and a ‘privileged direction’ arg ζ = 0. Points on or over ℂ_{•} will be denoted by ζ. Endless continuability for a (ramified) analytic germ φ̂ at 0_{•} means lateral analytic continuability along any broken line Γ on ℂ_{•} starting from 0_{•} under right and left circumvention of a discrete point set Sing_{φ̂, Γ} ⊂ Γ.

12This algebra underlies for ex. the construction of the trigebra of analysable germs.

Resurgent functions are subject to the convolution product, and there act upon them an incredibly rich array of alien derivations, which measure their singular behaviour away from 0_{•}. The label resurgent owes its origin to the fact that, at least in natural situations, alien derivatives tend to ‘resemble’, or be simply related to, the original function which thus ‘resurfaces’ at its singular points.

Local aspect: minors/majors :

More precisely, locally at 0_{•}, resurgent functions may be thought of as microfunctions: they are pairs φ^{⋄} (ζ) = { φ̌(ζ) , φ̂(ζ) } consisting of a major φ̌ defined upto a regular germ, and of a minor φ̂, which is the “variation” of the major. Depending on the situation, one may find it easier to work with real-majors and natural-majors

φ̂(ζ) = −\frac{1}{2πi} (φ̌_{real}(e^{πi}ζ) − φ̌_{real}(e^{−πi}ζ))  (ζ close to 0_{•}) (39)
φ̂(ζ) = φ̌_{nat}(ζ) − φ̌_{nat}(e^{−2πi}ζ)  (ζ close to 0_{•}) (40)
φ̌_{real}(ζ) ≡ 2πi φ̌_{nat}(e^{−πi}ζ)  (ζ close to 0_{•}) (41)

Real-majors are particularly recommended when dealing with acceleration theory and, of course, with real resurgent functions : they make it possible to associate real majors φ̌ to any real φ^{⋄}. Unless otherwise specified, all formulae below use real-majors.

The minor is exactly defined, but does not encapsulate the whole information about φ^{⋄}, except in the important case of “integrable resurgent functions”. For these the convolution product is defined by

(φ̂_{1} ⋆ φ̂_{2})(ζ) := ∫_{0}^{ζ} φ̂_{1}(ζ_{1}) φ̂_{2}(ζ − ζ_{1}) dζ_{1}  (42)

for ζ close to +0 and by analytic or cohesive continuation in the large. The convolution for general germs φ^{⋄} is defined by :

φ^{⋄}_{3} = φ^{⋄}_{1} ⋆ φ^{⋄}_{2} ⟺  (43)
φ̌_{3,u}(ζ) = \frac{1}{2πi} ∫_{I_{ζ,u}} φ̌_{1}(ζ_{1}) φ̌_{2}(ζ − ζ_{1}) dζ_{1} (0 < ζ < u << 1) (44)

with integration on the interval

I(ζ, u) := [\frac{1}{2} ζ + e^{− \frac{πi}{2}} u, \frac{1}{2} ζ + e^{+ \frac{πi}{2}} u]  (45)

The above definition makes sense since, upto addition of a regular germ, the major φ̌_{3,u} doesn’t depend on the choice of u. Moreover, for integrable functions φ^{⋄}, the two definitions (42) and (44) coincide.

Global aspect: alien derivations :

What really matters, however, is not the local or microfunction aspect, but the global properties of resurgent functions φ^{⋄}, which come from their having endlessly continuable, but usually highly ramified minors φ̂. This a source of many fascinating developments, chief amongst which is the existence of a system {Δ_{ω} , ω ∈ ℝ^{+}} of so-called alien derivations^{13}, which measure the singularities of the minor φ̂ over the points ω and which, together, freely generate an infinite dimensional Lie algebra with a non-countable basis – despite their acting on functions of one single variable !

The three models: formal, geometric, convolutive :

Actually, resurgent functions exist simultaneously in three different models :

– the formal model, with decreasing power series φ̃(z) (or suitable generalisations, like the so-called transseries) as elements, and with formal multiplication as product

– the convolutive model, defined a moment ago, with endlessly continuable microfunctions φ^{⋄} (ζ) = {φ̂(ζ), φ̌(ζ)} as elements, and with local-global convolution as product

– the geometric model, with sectorial analytic germs φ(z) as elements and with point-wise multiplication as product.

Roughly speaking, the way things work out in practice is like this : when formally solving singular equations or systems, we tend to get divergent solutions φ̃(z) of resurgent type, which do not converge directly to φ(z), but have to be summed in a round-about way, via the convolutive model φ^{⋄} (ζ). The detour, however, is rewarding, since it yields a treasure-trove of information about the equation, in particular its analytic invariants.

geometric model φ (z) Laplace
↑  ↘
⋮  φ^{⋄} (ζ) convolutive model
↑  ↗
formal model  φ̃ (z) Borel

13because they are indeed derivations, relative to the convolution product.

Standard Borel transform : φ(z) → φ^{⋄} (ζ) = (φ̂(ζ), φ̌(ζ))

φ̂(ζ) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} exp(zζ) φ(z) dz  (1 ≪ c ; arg ζ = 0)  (46)
φ̌(ζ) = ∫_{+u}^{+∞} exp(−zζ) φ(z) dz  (1 ≪ u ; | arg ζ| ≤ π) (47)
Standard Laplace transform :  φ^{⋄} (ζ) = (φ̂(ζ), φ̌(ζ)) → φ(z)
φ(z) = ∫_{+0}^{+∞} exp(−zζ) φ̂(ζ) dζ  (for φ^{⋄} integrable at 0_{•}) (48)
φ(z) = \frac{1}{2πi} ∫_{e^{−πi}∞}^{e^{πi}∞} exp(zζ) φ̌(ζ) dζ  (for any φ^{⋄} ; arg z = 0) (49)

Elementary (standard) Borel/Laplace transforms :

φ(z) = z^{−σ}  (for σ ∈ ℂ − ℕ^{⋆})
φ̂(ζ) = ζ^{σ−1}/Γ(σ)
φ̌(ζ) = ζ^{σ−1}Γ(1 − σ)  (50)
φ(z) = z^{−n}  (for n ∈ ℕ^{⋆})
φ̂(ζ) = ζ^{n−1}/Γ(n)
φ̌(ζ) = (−1)^{n} ζ^{n−1} log ζ /Γ(n)  (51)

2.3 Alien derivations or automorphisms. Their weights.

Alongside the natural derivation ∂ := ∂_{z} in the multiplicative (ie formal or geometric) models and its counterpart ˆ∂: φ̂(ζ) ↦ − ζ φ̂(ζ) in the convolutive model, resurgent functions are acted upon by so-called alien derivations Δ_{ω}. These are primarily defined in the convolutive model:

Δ_{ω} : φ^{⋄}:= (φ̂, φ̌) ↦ φ^{⋄}_{ω}= (φ̂_{ω}, φ̌_{ω})  (52)

where they measure the singularities of minors over^{14} a given singular point ω in ℝ^{+} or ℂ_{•} ^{15} but in such a way as to verify the Leibniz rule :

Δ_{ω} (φ^{⋄}_{1} ⋆ φ^{⋄}_{2}) ≡ Δ_{ω} (φ^{⋄}_{1}) ⋆ φ^{⋄}_{2} + φ^{⋄}_{1} ⋆ Δ_{ω} (φ^{⋄}_{2})  (53)

14 ie at various points lying over ω on various Riemann leaves.

15depending on whether we are dealing with real or general resurgent functions.

As a microfunction, φ^{⋄}_{ω} is defined by :

φ̂_{ω} (ζ) := + ∑_{ϵ_{i}=±} d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} φ̂^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})}(ω + ζ)  (54)
φ̌_{ω} (ζ) := − ∑_{ϵ_{i}=±} d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{r−1}, \atop ω_{r−1},} {∗ \atop ωr})} φ̂^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{r−1} \atop ω_{r−1}} )}(ω − ζ) (55)
d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{r−1}, \atop ω_{r−1},} {∗ \atop ωr})} := +2πi d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{r−1}, \atop ω_{r−1},} {+ \atop ωr})} = −2πi d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{r−1}, \atop ω_{r−1},} {− \atop ωr})}

for arg ζ = arg ω_{i} = arg ω and ζ small^{16}. Here, of course, φ̂^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})}(ω + ζ) denotes the determination of φ̂(ζ) on the open interval ]ω_{r}, ω_{r+1}[ that answers to the right (resp left) circumvention of ω̂_{i} ^{17} to the right (resp left) if ϵ_{i} = + (resp −). Due to (56), φ^{⋄}_{ω}= 0 if φ̂ has no singularities over ω. Moreover, since the right-hand side of (54) or (55) should not change when additional, ficticious singular points ω̂_{i} are added, the weights d^{•} are subject to obvious self-consistency relations (164) : see §6.1.

The simplest choice for “singularity measuring” operators would be the right-lateral operators Δ_{ω}^{+} , with weights :

d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := ϵ_{r} 1 if (ϵ_{1}, …, ϵ_{r−1}) = (+, …, +)
:= 0  otherwise

and the left-lateral operators Δ_{ω}^{−}, with weights :

d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := − ϵ_{r} 1 if (ϵ_{1}, …, ϵ_{r−1}) = (−, …, −)
:= 0  otherwise

But the “atomicity” of these operators (ie the fact that they involve one path only) isn’t much of an asset, because they are not first-order derivations. Instead of verifying the Leibniz rule (53), they verify messier relations, which for simplicity we write down in the multiplicative models :

Δ_{ω}^{±}(φ_{1} φ_{2}) ≡ (Δ_{ω}^{±}φ_{1}) φ_{2} + φ_{1} (Δ_{ω}^{±}φ_{2}) + ∑_{{ω_{1}+ω_{2}=ω \atop arg ω_{i}=arg ω}}(Δ_{ω_{1}}^{±}φ_{1}) (Δ_{ω_{2}}^{±}φ_{2}) (56)

with a sum extending to all ω_{1}, ω_{2} co-axial with ω. ^{18}

16analytic continuation extends these definitions for large ζ.

17ω̂_{i} being of the i-th singular point ω̂_{i} := ∑_{1}^{i} ω_{k} of φ̂ lying on the interval ]0_{•}, ω[.

18For any given pair of test functions φ_{1}, φ_{2} the above sum makes sense, since it can never involve more than a finite number of non-zero terms.

The so-called standard alien derivations Δ_{ω} do verify (53), but they involve all the 2^{r} determinations which forward analytic continuation may create over [ω_{r}, ω_{r+1}]. Still, their weights are very simple, as they do not depend on the gaps ω_{i}, but only on the number p, q of +, − signs in the sequence {ϵ_{1}, …, ϵ_{r−1}}:

d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := \frac{ϵ_{r}}{2πi} \frac{p! q!}{(p + q + 1)!}  {p := #\{1 ≤ i < r; ϵ_{i} = +\} \atop q := #\{1 ≤ i < r; ϵ_{i} = −\}} (57)

Together, these standard^{19} derivations Δ_{ω} freely generate the algebra ALIEN of alien derivations but, as we shall see later on, their simplicity appeal is deceptive, and in many applications, preference should be given to another system, the so-called organic alien derivations.

There being no risk of confusion, the same symbols Δ_{ω} serve to denote the alien derivations in all three models – convolutive (where they are defined and where only they can be directly calculated), formal, and geometric.

In the two later models (collectively referred to as multiplicative models) it is also useful to consider the exponential-carrying variants Δ Δ_{ω} := e^{−ωz}Δ_{ω} which have the advantage of commuting with the natural derivation ∂ := ∂_{z} and of enjoying the invariance property (9), but at the price of introducing an exponential factor external to the algebra of resurgent fonctions.

2.4 Resurgence monomials.

They are basically systems of resurgent fuctions that

(1) are as elementary as possible and, ideally, canonical

(2) are easy to multiply or convolute^{20}

(3) behave simply under natural differentiation (3′) behave simply under alien differentiation

(4) are numerous enough to generate the resurgent solutions of singular equations – be these of differential, difference, functional, etc, type (4′) are numerous enough to permit the reconstitution of any resurgent function with a prescribed resurgence pattern, ie verifying a prescribed system of resurgence equations.

19 usually, the standard alien derivations are defined without the factor \frac{1}{2πi} , so as to give them real weights. But in this investigation, more important than having real weights is the property for an operator of turning a real resurgent function into another real resurgent function. In view of the self-consistency relation (165), this makes the introduction of the factor \frac{1}{2πi} necessary.

20depending on the model we happen to be working in. Here, for notational convenience, we shall assume it to be the formal model

Such systems of monomials do exist, but there is a tension between the demands (3),(4) and (3′),(4′).

If we give precedence to (3),(4), we get the so-called ∂-friendly resurgence monomials V^{•}(z), which are particularly useful in all problems of “Object Analysis” (object ↦ invariants)

If we give precedence to (3′),(4′), we get the so-called Δ-friendly resurgence monomials U^{•}(z), which are particularly useful in all problems of “Object Synthesis” (invariants ↦ object)

The alien derivatives Δ_{ω}V^{•}(z) of the ∂-friendly monomials involve simpler ∂-friendly monomials, but also transcendental constants V^{•}, the so-called ∂- friendly monics, which measure as it were the Δ-unfriendliness of the V^{•}(z).

Similarly, the natural derivatives ∂_{z} U^{•}(z) of the Δ-friendly monomials involve simpler Δ-friendly monomials, but also transcendental constants U^{•}, the so-called Δ-friendly monics, which so to speak measure the ∂-unfriendliness of the U^{•}(z).

The ∂-friendly monics make it possible to express the analytic invariants 𝔸_{ω} of a local object Ob in terms of its Taylor coefficients.

The Δ-friendly monics, or at any rate particularly well-chosen systems of such monics, make the reverse passage possible : expressing the Taylor coefficients of a (canonical) object Ob^{can} in terms of a prescribed system of invariants 𝔸_{ω}.

The monics also serve as a bridge between the two types of monomials : see (65) below.

Lastly, just as there exist two useful variants of alien derivations, the exponential-free Δ_{ω} and the exponential-carrying Δ Δ_{ω}, we have to consider, alonside the exponential-free resurgence monomials, the exponential-carrying variants, but with the opposite sign conventions (for ‘orthogonality’) :

Δ Δ_{ω} := e^{−ω z} Δ_{ω}  (58)
Ve^{ω}(z) := e^{+∥ω∥ z} V^{ω}(z)  (59)
Ue^{ω}(z) := e^{+∥ω∥ z} U^{ω}(z)  (60)

To make the above statements sound a little less abstract, let us construct four very elementary systems of resurgence monomials and monics. They are often referred to as hyperlogarithmic, for such indeed is their dependence on the indices ω_{i} and (in the convolutive model) on the variable ζ. The construction involves four steps :

Ve^{•}(z) ⟶ V^{•} ⟶ U^{•} ⟶ Ue^{•}(z)  (61)

and the definitions, relative to the formal model, go as follows :

∂_{z}Ṽe^{ω_{1},...,ω_{r}}(z) := − Ṽe^{ω_{1},...,ω_{r−1}}(z) e^{ω_{r} z} z^{−1}  (62)
Δ_{ω_{0}} Ṽe^{ω_{1},...,ω_{r}}(z) =: ∑_{ω_{1}+···+ω_{i}=ω_{0}} V ^{ω_{1},...,ω_{i}} Ṽe^{ω_{i+1},...,ω_{r}}(z)  (63)
U^{•} := inverse of V^{•} under mould-composition ∘ (64)
Ũe^{•}(z) := Ṽe^{•}(z) ∘ U^{•}  (65)

Clearly^{21} :

Ṽe^{ω_{1},...,ω_{r}}(z) ∈ e^{(ω_{1}+···+ω_{r}) z} ℂ[[z^{−1}]]  (66)
Ũe^{ω_{1},...,ω_{r}}(z) ∈ e^{(ω_{1}+···+ω_{r}) z} ℂ[[z^{−1}]]  (67)

The above systems of ∂-friendly monomials and monics suffice for object analysis in the monocritical examples 1, 2, 3, 4 of the next section. In fact, larger systems, with the one-index factor e^{ω_{r} z} z^{−1} in (62) replaced by the two-index factor e^{ω_{r} z} z^{−1−σ_{r}} , suffice for most monocritical examples. Lastly, by considering factors of type e^{P_{r}(z)} z^{−1} (resp e^{P_{r}(z)} z^{−1−σ_{r}}), with a general polynomial P_{r}(z) inside the exponential, one can construct systems large enough to deal with object analysis in our polycritical examples 5 and 6 (resp in most conceivable polycritical examples). In none of these instances, however, are the corresponding Δ-frienly systems adequate to the job of object synthesis : as we shall see, two extra ingredients – a twist parameter along with an antipodal symmetry – are called for to make things click.

3 Object Analysis: six basic examples.

In the following six examples, we consider analytic local objects Ob (diffeos, vector fields, differential equations or sustems) which we assume to be formally conjugate to a normal form Ob^{nor}. In each instance, we choose the simplest formal class that is apt to illustrate the particular point of analysis we wish to make.

3.1 Example 1: shift-like diffeomorphism.

f^{nor} : z ⟶ z + 2πi  (68)
f  : z ⟶ z + 2πi + ∑_{2≤n} a_{n} z^{−n}  (69)

21 in theory the indices ω_{i} are in ℂ_{•} but here they can be assumed to be in ℂ since the minors of our resurgent functions (in all six examples) are not ramified at 0_{•} ∈ ℂ_{•} .

Remark : we might of course have chosen the unit shift as our normal form, but choosing the 2πi-shift has the advantage of placing the singularities over ℤ in the Borel-plane, and of rendering the parallel with Example 2 (infra) more obvious.

We may work with the formal normalising map f^{⋆} or its inverse ^{⋆}f :

f^{⋆} ∘ f ≡ f^{nor} ∘ f^{⋆}  ie  f^{⋆}(f(z)) ≡ 2πi + f^{⋆}(z)  (70)
f ∘ ^{⋆}f ≡ ^{⋆}f ∘ f^{nor}  ie  f(^{⋆}f(z)) ≡ ^{⋆}f(z + 2πi)  (71)

Both are generically divergent but always resurgent. They verify the resurgence equations :

Δ Δ_{n} f^{⋆}(z) ≡ −A_{n} exp(−n f^{⋆}(z))  (∀n ∈ ℤ^{⋆}) (72)
Δ Δ_{n} ^{⋆}f(z) ≡ +A_{n} e^{−nz}∂_{z} ^{⋆}f(z) =: 𝔸_{n} ^{⋆}f(z)  (∀n ∈ ℤ^{⋆}) (73)

which in turn yield the complete and ‘free’ ^{22} system of analytic invariants :

𝔸 = {𝔸_{n} := A_{n} e^{−nz}∂_{z} ; n ∈ ℤ^{⋆} , A_{n} ∈ ℂ}  (74)

For details see [E2],[E3].

3.2 Example 2: Euler-like differential equation.

Example 2: singular, non-linear differential equation.^{23}

d_{z} y^{nor} = y^{nor}  (75)
d_{z} y  = y + ∑_{1+n≥0} b_{n}(z) y^{1+n}  ∈ y + ℂ{y, z^{−1}}  (76)

We may work with the formal integral :

y(z, u) ∈ ℂ[[z^{−1}, u e^{z}]]  (u = integration parameter) (77)

which is generically divergent (in z) but always resurgent (again, in z) and verifies the Bridge equation :

Δ Δ_{n} y(z, u) ≡ A_{n} u^{n+1} ∂_{u} y(z, u) =: 𝔸_{n} y(z, u)  (n = −1, 1, 2, 3, …) (78)

yielding the complete and ‘free’ system of analytic invariants :

𝔸 = {𝔸_{n} := A_{n} u^{n+1} ∂_{u} ; n ∈ {−1} ∪ ℕ^{⋆} , A_{n} ∈ ℂ}  (79)

For details, see [E3].

22see the remark at the end of §3.

23under addition of a variable x := z^{−1} it translates into a two-dimensional vector field, local (at 0 ∈ ℂ^{2}), singular, and resonant (with one vanishing and one non-vanishing multiplier)

3.3 Example 3: monocritical linear differential system.

d_{z} y_{i}^{nor} = λ_{i} y_{i}^{nor}  (1 ≤ i ≤ ν ; λ_{i} ≠ λ_{j} if i ≠ j) (80)
d_{z} y_{i}  = λ_{i} y_{i} + ∑_{1≤j≤ν} b_{i,j}(z) y_{j}  b_{i,j}(z) ∈ ℂ{z^{−1}} (81)

Here the formal integral reduces to

y(z, u) = ∑_{1≤i≤ν} b_{i}(z) e^{λ_{i} z} u_{i}  with b_{i}(z) ∈ ℂ[[z^{−1}]]  (82)

The Bridge equation reads :

Δ Δ_{λ_{i}−λ_{j}} y(z, u) ≡ A_{λ_{i}−λ_{j}} u_{i} ∂_{u_{j}} y(z, u) =: 𝔸_{λ_{i}−λ_{j}} y(z, u) (i ≠ j) (83)

and once again yields a complete and free, but this time finite, system of analytic invariants :

{𝔸_{λ_{i}−λ_{j}} := A_{λ_{i}−λ_{j}} u_{i} ∂_{u_{j}} ; 1 ≤ i ≠ j ≤ ν}  (84)

For details, see [E3].

3.4 Example 4: monocritical non-linear differential system.

d_{z} y_{i}^{nor} = λ_{i} y_{i}^{nor}  (1 ≤ i ≤ ν ; λ not res. nor quasi.res.) (85)
d_{z} y_{i}  = λ_{i} y_{i} + ∑_{{1+n_{i}≥0 \atop n_{j} ≥0 if j≠i}} b_{i,n}(z) y_{i} y^{n} ∈ λ_{i} y_{i} + ℂ\{z^{−1}, y_{1}, …, y_{ν}\}(86)

The formal integral involves ν integration parameters u_{i}, each with its accompanying exponential factor :

y(z, u) ∈ ℂ[[z^{−1}, u_{1} e^{λ_{1} z}, …, u_{ν} e^{λ_{ν} z}]] (with ℚ-independent λ_{i}’s) (87)

The Bridge equation reads :

Δ Δ_{ω} y(z, u) ≡ 𝔸_{ω} y(z, u)  (∀ω = Ω)  (88)

with indices ω running through a set :

Ω = {ω ; ω = ∑_{1≤i≤ν} m_{i} λ_{i} , m_{i} ≥ −1 , ∑_{m_{i}=−1} 1 = 0 or 1}  (89)

and with differential operators of the form :

𝔸_{ω} := u_{1}^{m_{1}} … u_{ν}^{m_{ν}} ∑_{1≤i≤ν} A_{ω}^{i} u_{i} ∂_{u_{i}}  if ω = ∑ m_{i} λ_{i} (A_{ω}^{i} ∈ ℂ) (90)

which, together, constitute a complete and free system {𝔸_{ω} ; ω = Ω} of analytic invariants. For details, see [E3].

3.5 Example 5: polycritical linear differential system.

p_{i}^{−1} t^{1+p_{i}} d_{t} y_{i}^{nor} + λ_{i} y_{i}^{nor} = 0  (1 ≤ i ≤ ν)  (91)
p_{i}^{−1} t^{1+p_{i}} d_{t} y_{i} + λ_{i} y_{i} = ∑_{1≤j≤ν} b_{i}(t) y_{j} (b_{i}(t) ∈ ℂ{t}) (92)

The Bridge equation reads :

Δ Δ_{ϖ} y(z, u) ≡ 𝔸_{ϖ} y(z, u)  (∀ω = Ω)  (93)

Here the alien derivations carry indices ϖ = ({ω \atop q}), with a lower q ∈ \{p_{1}, …, p_{s}\} signalling the critical variable z_{q} := t^{−q} respective to which the alien derivation operates, and with an upper ω of the form λ_{i} − λ_{j} or λ_{i} or λ_{j} for p_{i} = p_{j} = q. The corresponding operators 𝔸_{ϖ} read:

\begin{aligned}𝔸_{ϖ} & := & A_{ϖ} u_{i} ∂_{u_{j}} & if & ϖ=( {λ_{i}−λ_{j} \atop q} ) \\ 𝔸_{ϖ} & := & u_{i} (∑_{p_{k}<q} A_{ϖ}^{k} ∂_{u_{k}}) & if & ϖ=( {λ_{i} \atop q}) \\ 𝔸_{ϖ} & := & (∑_{p_{k}<q} A_{ϖ}^{k} u_{k} )∂_{u_{j}} & if & ϖ=( {−λ_{j} \atop q} ) \end{aligned}

As usual, the Bridge Equation yields a complete and free but (due to linearity) finite system of analytic invariants :

{A_{λ_{i}−λ_{j}} ; p_{i} = p_{j}} ∪ {A_{λ_{i}}^{k} ; p_{i} > p_{k}} ∪ {A_{−λ_{j}}^{k} ; p_{j} > p_{k}}  (94)

For details, see [E3],[E4],[E7].

3.6 Example 6: polycritical non-linear differential system.

p_{i}^{−1} t^{1+p_{i}} d_{t} y_{i}^{nor} + λ_{i} y_{i}^{nor} = 0  (1 ≤ i ≤ ν) (95)
p_{i}^{−1} t^{1+p_{i}} d_{t} y_{i} + λ_{i} y_{i} = b_{i}(t, y_{1}, …, y_{r}) ∈ ℂ{t, y}  (96)

The Bridge equation reads :

Δ Δ_{ϖ} y(z, u) ≡ 𝔸_{ϖ} y(z, u)  (∀ω = Ω)  (97)

Here the alien derivations carry indices ϖ = ({ω \atop q}), with ω running through a set :

Ω_{q} = {ω ; ω = ∑_{p_{i}=q} m_{i} λ_{i} , m_{i} ≥ −1 , ∑_{m_{i}=−1} 1 = 0 or 1}  (98)

and with differential operators of the form :

𝔸_{ϖ} := u^{n(ϖ)} {∑_{p_{j}≥q} A_{ϖ}^{j}(u) u_{j} ∂_{u_{j}} + ∑_{p_{j}<q} A_{ϖ}^{j}(u) ∂_{u_{j}}} with (99)
u^{n(ϖ)} := ∏_{p_{i}=q} u_{i}^{m_{i}} if ω = ∑ m_{i} λ_{i}  (100)
A_{ϖ}^{j}(u) ∈ ℂ[[u_{j}; p_{j} < q]])  (101)

which, together, constitute a complete and ‘free’ system {𝔸_{ω} ; ω = Ω} of analytic invariants. For details, see [E3],[E4],[E7],[Braa] .

Remark : Of course, in our four non-linear examples, the countably infinite systems of invariants {𝔸_{ω}} are “free” only in the sense of being subject to no finite constraints, ie constraints involving finite subsets of them. But they are clearly subject to ‘infinite’ constraints which, relative to a nice (“wellbehaved”, see infra) basis of ALIEN, reduce to the existence of exponential bounds in ω.

4 The reverse problem: Object Synthesis.

In this section, we examine what happens when we try to perform canonical Object Synthesis with the elementary, hyperlogarithmic resurgence monomials (constructed at the end of §2.4) as our sole tool : in some very simple cases the method works; in most others it fails. Yet these failures are instructive, for they help identify the two main difficulties and already suggest the correct remedy, namely the introduction of a well-placed ‘twist parameter’.

4.1 Standard or hyperlogarithmic resurgence monomials and monics.

Let us at the oustet collect a few formulas about the Δ-friendly monomials and monics :

∂−friendly  Δ−friendly
monomials  V^{•}(z)  ⟷  U^{•}(z)
↓  ↖ ↗  ↑
monics  V^{•}  ⟷  U^{•}

U^{ω′}(z) U^{ω′′}(z) = ∑_{ω∈sha(ω′,ω′′)} U^{ω}(z)  (102)

∂_{z} U^{ω_{1},...,ω_{r}}(z) = − ∑_{1≤i≤r} U^{ω_{1},...,ω_{i}}(z) U^{ω_{i+1},...,ω_{r}} z^{−1}  (103)
Δ_{ω_{0}} U^{ω_{1},...,ω_{r}}(z) = U^{ω_{2},...,ω_{r}}(z) if ω_{0} = ω_{1}  (104)
= 0  if ω_{0} ≠ ω_{1}  (105)

Let us also mention an integral representation capable of a treble interpretation – in the formal, geometric, and convolutive models ^{24} :

U^{ω_{1},...,ω_{r}}(z) := SPA ∫_{0}^{∞} \frac{exp(−ω_{1}y_{1} ⋯ − ω_{r}y_{r})}{(y_{r} − y_{r−1}) … (y_{2} − y_{1})(y_{1} − z)} dy_{1} … dy_{r} (106)

4.2 Semi-formal synthesis in Example 1.

Recall that in this example each invariant reduces to a single scalar :

𝔸_{n} := A_{n} e^{−nz} ∂_{z}  (107)

So we may start from some admissible system {… A_{−2}, A_{−1}, A_{1}, A_{2} … } and set:

F^{⋆} := 1 + ∑_{r} ∑_{n_{i}}(−1)^{r} Ue^{n_{1},n_{2},...,n_{r}}(z) 𝔸_{n_{r}} … 𝔸_{n_{2}}𝔸_{n_{1}}  (108)

24for the interpretation in the various models and the meaning of SPA – special path averaging – see also §10.3 infra.

Simple algebra^{25} shows that F^{⋆} is a formal^{26} automorphism of ℂ[[z^{−1}]]:

F^{⋆} (φ_{1} ∗ φ_{2}) ≡ (F^{⋆} φ_{1}) ∗ (F^{⋆} φ_{2})  ∀ φ_{1}, φ_{2}  (109)

and therefore a substitution operator of the form F^{⋆} φ ≡ φ ∘ f^{⋆} with^{27} :

f^{⋆}(z) := z − ∑_{r} ∑_{n_{i}} U^{n_{1},...,n_{r}}(z) Γ_{n_{1},...,n_{r}} A_{n_{1}} … A_{n_{r}}  (110)
Γ_{n_{1}} := 1  (111)
Γ_{n_{1},...,n_{r}} := n_{1}(n_{1} + n_{2}) … (n_{1} + … n_{r−1}) (r ≥ 1)  (112)

Then f^{⋆} formally verifies the system of resurgence equations :

Δ Δ_{n}f^{⋆}(z) = −A_{n} exp(−nf^{⋆}(z)) (∀n)  (113)
Δ_{n}f^{⋆}(z) = −A_{n} exp(−n (f^{⋆}(z) − z)) (∀n)  (114)

Therefore, if we set

f(z) := f^{⋆}(2πi + ^{⋆}f(z))  (115)

and apply the rule for alien-differentiating a composition product :

Δ Δ_{ω}(g ∘ h) ≡ (Δ Δ_{ω}g) ∘ h + (∂_{z} g) ∘ h . Δ Δ_{ω}h (if h(z) ∼ z)  (116)

we find that all alien derivatives of f are identically zero. So f is “formally” an identity-tangent analytic diffeo with the prescribed invariants.

4.3 Semi-formal synthesis in Example 2.

Here again, the invariants 𝔸_{n} := A_{n} u^{n+1} ∂_{u} reduce to a single scalar A_{n}, but they no longer depend on z and their index n can no longer assume negative values other than the exceptional −1. So let us start from some admissible system {A_{−1}, A_{1}, A_{2}, A_{3}, … } and set:

Θ := 1 + ∑_{r} ∑_{n_{i}} (−1)^{r} Ue^{n_{1},n_{2},...,n_{r}}(z) 𝔸_{n_{r}} … 𝔸_{n_{2}}𝔸_{n_{1}} (117)
Θ^{−1} := 1 + ∑_{r} ∑_{n_{i}}  Ue^{n_{r},...,n_{2},n_{1}}(z) 𝔸_{n_{r}} … 𝔸_{n_{2}}𝔸_{n_{1}} (118)

25using the multiplication rule (102) for the U^{•}.

26but, to get lighter notations, we drop the tildas

27mark the replacement of Ue^{•} by U^{•} and the non-dependence of Γ_{n_{1},...,n_{r}} on n_{r}.

Due once again to (102), the operatorrs Θ^{±1} are formal algebra automorphisms or, more precisely, substitution operators :

Θ^{±1}(φ_{1} φ_{2}) ≡ Θ^{±1}(φ_{1}) Θ^{±1}(φ_{2})  (119)

This standard procedure, when applied with respect to the hyperlogarithmic resurgence monomials mentioned at the end of §2.4, leads from an admissible system of prescribed invariants:

{A_{−1}, A_{1}, A_{2}, A_{3}, … }  (120)

to an singular differential equation of the form:

d_{z}y = y − z^{−1}{B_{−1} + B_{1}y^{2} + B_{2}y^{3} + B_{3}y^{4} + … }  (121)

with a perfect symmetry of form between the invariants {A_{n}} of our “object” and its Taylor coefficients {B_{n}}. Indeed, if we introduce operators 𝔹_{n} := B_{n} u^{n+1}∂_{u} parallel to the operators 𝔸_{n} := A_{n} u^{n+1}∂_{u} and then plug into (117),(118) the formula (103) that express the natural derivatives of the Δfriendly monomials U^{•}(z), setting aside for the time being all questions of convergence, we find the following, engagingly symmetric formulae for the correspondance {A_{n}} ↔ {B_{n}} :

𝔹_{n_{0}} = ∑_{r≥1} ∑_{n_{i}} U^{n_{1},...,n_{r}} 𝔸_{n_{r}} … 𝔸_{n_{2}} 𝔸_{n_{1}} (122)

(due to alternality) = ∑_{r≥1} \frac{1}{r} ∑_{n_{i}} U^{n_{1},...,n_{r}}[𝔸_{n_{r}} … [𝔸_{n_{2}}, 𝔸_{n_{1}}]] (123)
𝔸_{n_{0}} = ∑_{r≥1} ∑_{n_{i}} V^{n_{1},...,n_{r}} 𝔹_{n_{r}} … 𝔹_{n_{2}} 𝔹_{n_{1}} (124)

(due to alternality) = ∑_{r≥1} \frac{1}{r} ∑_{n_{i}} V^{n_{1},...,n_{r}}[𝔹_{n_{r}} … [𝔹_{n_{2}}, 𝔹_{n_{1}}]] (125)

4.4 Semi-formal synthesis in the remaining examples.

For Example 3 and 4, the procedure is exactly the same as above, in §4.2, for Example 2 :

𝔸_{ω} := u^{n+1} ∑ A_{ω}^{j} u_{j}∂_{u_{j}}  (126)
Θ := 1 + ∑_{r} ∑_{ω_{i}}(−1)^{r}Ue^{ω_{1},ω_{2},...,ω_{r}}(z) 𝔸_{ω_{r}} … 𝔸_{ω_{2}}𝔸_{ω_{1}}  (127)
Θ^{−1} := 1 + ∑_{r} ∑_{ω_{i}} Ue^{ω_{r},...,ω_{2},ω_{1}}(z) 𝔸_{ω_{r}} … 𝔸_{ω_{2}}𝔸_{ω_{1}}  (128)

except for the required modifications in the size of the index reservoir Ω (see (83),(89)) and the shape of the invariants 𝔸_{ω} (see (84),(90)).

For the polycritical examples 5 (linear) and 6 (non-linear), the changes are more thorough-going :

– due to the need to apply acceleration transforms^{28} for jumping from one critical time to the next,

– due to the need to choose a specific integration axis θ_{i} in each critical Borel plane, which results in a general multipolarisation that affects not only the geometric model, but also the resurgence monomials and the very invariants 𝔸_{ω}, except of course those attached to the ‘first’ or ‘slowest’ critical time,

– due to the presence in the invariants 𝔸_{ω} of parameters u_{j} attached to the ‘earlier’ or ‘slower’ critical times, with the result that each 𝔸_{ω} may now carry (in the non-linear Example 6) infinitely many scalar coefficients.

But in all these cases, hyperlogarithmic object synthesis displays one striking feature, already apparent in the formulas (122),(123) and (124),(125), namely a rigorous (at least at the formal level) duality between the invariants and the Taylor coefficients of the canonical object Ob^{can} produced by synthesis. The natural indexation, in particular, is exactly the same – whereas for generic objects Ob in a given conjugacy class the Taylor coefficients tend to be “infinitely more numerous”^{29} than the invariants. This extremely nice feature of semi-formal, hyperlogarithmic synthesis will get lost, alas, when we go over to effective, paralogarithmic synthesis but it won’t disappear altogether : the Taylor coefficients of the new Ob^{can} will be exactly twice as numerous (no more !) than its invariants.

4.5 Inadequacy of the standard resurgence monomials for the purpose of synthesis.

In Example 1 and 2 alike, the analytic invariants 𝔸_{n} and the operators 𝔹_{n} that encode the Taylor-coefficients^{30} of Ob^{can} essentially reduce to scalars A_{n} , B_{n}. Looking at the (formally inverse) relations (122-125) which exchange them, we are led to distinguish exactly five cases :

28see [E4],[E7].

29not in terms of cardinality, of course, which is always countable, but in terms of natural indexability by sets of type ℕ^{m} × ℤ^{n}.

30directly so in Example 2; indirectly so, via equation (115), in Example 1.

4.5.1 Unary case:

All {A_{n} , B_{n}} vanish, save for one value of the index, n = n_{0}, which, due to scale-invariance^{31} , we may take to be 1. Then the relations (122-125) immediately give :

{A_{1}} ⟺ {B_{1}} with A_{1} = B_{1}  (129)

4.5.2 Binary case:

All {A_{n} , B_{n}} vanish, save for two opposite values of the index, n = ±n_{0}, which, due to scale-invariance, we may take to be ±1. After some calculations (see [E2]), the relations (122-125) yield :

{A_{−1}, A_{1}} ⟵ {B_{−1}, B_{1}}  (130)
with  \frac{1}{2}\sqrt{A_{−1} A_{1}} = sin (\frac{1}{2}\sqrt{B_{−1} B_{1}})  (131)
and  A_{1}/A_{−1} = B_{1}/B_{−1}  (132)

So, going from semi-formal to effective synthesis, we register the existence of countably many hyperlogarithmic Ob^{can}, with no clear privileged choice except for small values of the product A_{−1} A_{1}.

4.5.3 Unilateral case:

All {A_{n} , B_{n}} vanish, save for indices n that are positive multiples of some n_{0} , which we may take to be 1. Actually, to get true unilaterality^{32} we must assume not only that … A_{−3} = A_{−2} = A_{−1} = 0 but also that A_{1} and at least one other A_{n} with positive index n are ≠ 0. This is the most promising case because (unlike the unary case) it is nontrivial and (unlike with the binary case) there is at the semi-formal level a clear one-to-one correspondance:

{A_{1}, A_{2}, A_{3}, … } ⟷ {B_{1}, B_{2}, B_{3}, … }  (133)

since for any given n_{0} the right-hand side of the relations(122-125) reduces to a finite number of summands. Unfortunately, going from semi-formal to effective synthesis, it can be shown that for nearly all ^{33} admissible unilateral systems {A_{n}} of invariants, the corresponding sequence {B_{n}} of Taylor coefficients has Gevrey-1 growth ^{34} in n, which disqualifies the hyperlogarithmic Ob^{can} as an effective synthesis.

31U^{l ω_{1},...,l ω_{r}} ≡ U^{ω_{1},...,ω_{r}} ; V^{l ω_{1},...,l ω_{r}} ≡ V^{ω_{1},...,ω_{r}}

32ie to exclude the unary and binary cases.

33the exact criterion is mentioned in §4.6 below

4.5.4 Sesquilateral case:

All {A_{n} , B_{n}} vanish, save for n a positive multiple of some n_{0} and also for n = −n_{0} . Here again, we may take n_{0} to be 1. In fact, it is enough to assume that A_{n} ≠ 0 for n = ±1 and for at least one other positive n . In terms of effective hyperlogarithmic synthesis, the picture is much the same as in the binary case : there exists one priviledged solution Ob^{can} if A_{−1} A_{1} is small enough. Otherwise, there are several (countably many) solutions on an equal footing.

4.5.5 Bilateral case:

All {A_{n} , B_{n}} vanish, save for n a positive or negative multiple of n_{0}, which we may take to be 1. In fact, it is enough to assume that A_{n} ≠ 0 for at least two distinct positive values of n and two distinct negative values. For definiteness let us assume that A_{±1} ≠ 0, A_{±2} ≠ 0.

Then there exists one priviledged Ob^{can} if A_{−1} A_{1} is small enough and if the two sequences {A_{n} , n > 0} and {A_{n} , n < 0}, or rather the Fourier series π^{±} built from the related sequences {A′_{n} , n > 0} and {A′_{n} , n < 0} (see (140),(141) infra) in some right and left half-planes, verify a suitable “overlapping condition”, like the one mentioned in §5.1 in the context of non-canonical synthesis. Failing that condition, the picture becomes quite murky, with either a countably infinity of solutions, or none at all.

4.6 First intimations of “antipodality”:

Let us take a closer look at the unilateral case. Assume for simplicty that A_{1} ≠ 0, A_{2} = 0 ^{35} , which under (122-125) also implies that B_{1} ≠ 0, B_{2} = 0. Assume further that the invariants A_{n} are expressed relative to the system of standard alien derivations : assume, in other words, that the A_{n} are the constants occuring in (72-73) with Δ Δ_{n} defined as in (54),(57).

34ie 0 < lim sup \frac{1}{n} |B_{n}|^{1/n} < ∞

35No restrictions are imposed on A_{3}, A_{4} . . .. The condition A_{2} = 0 is simply for convenience (to remove the logarithms in A^{∗} and B^{∗}). Dropping it would make no difference to the analysis.

To describe the precise obstructions to effective hyperlogarithmic synthesis in complete, closed form, we require a fair number of auxiliary objects. First, we successively define six power series A_{∗} , A^{∗} , A and B_{∗} , B^{∗} , B:

\begin{aligned}A_{∗}(y) & := & ∑_{n≥1} A_{n} y^{n+1} & B_{∗}(y) & := & ∑_{n≥1} B_{n} y^{n+1} \\ A^{⋆}(y) & := & ∫ \frac{dy}{A(y)} & B^{⋆}(y) & := & ∫ \frac{dy}{B(y)} \\ A^{∗} ∘ A & ≡ & 2 πi + A^{∗} & B^{∗} ∘ B & ≡ & 2 πi + B^{∗} \end{aligned}

Further, we form the partial differential operator :

D := ∂_{z} + {y − z^{−1}B_{∗}(y)} ∂_{y}  (134)

We have to solve equation DK = 0 formally in decreasing powers of z. We also have to solve equation DQ = 0 formally in increasing powers of z. In both cases, we get increasing (but not always positive) powers of y and it is convenient to expand the solutions as follows :

K(z, y) := e^{−z} y + e^{−z} ∑_{n≥1} K_{n}(z) y^{n+1}  with K_{n}(z) ∈ ℂ[[z^{−1}]]
Q(z, y) := log z + B^{⋆}(y) + ∑_{m≥1} Q_{m}(y) z^{m} with B^{∗}(y), Q_{n}(y) ∈ y^{−1} ℂ[[y]]

Then we require three projectors P^{0}, P^{+}, P^{−} acting on series ∑_{−∞}^{+∞} c_{n} z^{−n}: P^{0} retains only the z-constants P^{+} retains only the positive powers of z P^{−} retains only the negative powers of z

Since we have expressed our {A_{n}} relative to the standard alien derivations, their quality of being an admissible system of invariants is not directly recognisable on A_{∗}(y) or A^{∗}(y), but rather on A(y). More precisely, {A_{n}} constitutes an admissible system of invariants iff A(y) is local-analytic ^{36}. As for A_{∗}(y) and A^{∗}(y)), being respectively the infinitesimal generator and normaliser of the local-analytic diffeo A, they are merely resurgent, with critical time s := −(2πi/A_{1}) y^{−1} and with the resurgence equations :

Δ Δ_{n}A^{⋆} = − C_{n} e^{−nA^{⋆}}  (n ∈ ℤ^{⋆} , Δ_{n} := Δ Δ_{n}^{{s}}) (135)

The constants C_{n} have a rather strange status : they are ‘invariants of invariants’ since the Taylor coefficients of the diffeo A are themselves rational functions of the invariants A_{n}. Yet they hold the key to our problem. Indeed, we are now in a position to state the main results of this section : as power series of y or Laurent series of s := const.y^{−1}, the series B_{∗} , B^{∗} , K , Q are generically divergent, but always resurgent, with critical time s, and they verify the system of resurgence equations :

36see [E3].

Δ Δ_{n}B^{⋆}/∂_{s}B^{⋆} = − C_{n} P^{0} (e^{−nQ}/∂_{s}Q)  (n ∈ ℕ^{⋆}) (136)
Δ Δ_{n}K/∂_{s}K = + C_{n} P^{−} (e^{−nQ}/∂_{s}Q)  (n ∈ ℕ^{⋆}) (137)
Δ Δ_{n}Q/∂_{s}Q = − C_{n} (P^{0} + P^{+}) (e^{−nQ}/∂_{s}Q)  (n ∈ ℕ^{⋆}) (138)

(139)

That system is complete and closed, in the sense that it enables us to calculate all the iterated alien derivatives Δ Δ_{n_{r}} … Δ Δ_{n_{1}} B^{∗} and Δ Δ_{n_{r}} … Δ Δ_{n_{1}} B_{∗} , whereas no smaller system could do that. But if we take a closer look at the above system, especially at (136), we notice that only the alien derivations Δ Δ_{n} with positive n are capable of acting non-trivially on B^{∗} and B_{∗}. Those with negative n yield 0, because for them the right-hand side of (136) vanishes. The upshot is that unilateral hyperlogarithmic synthesis is possible if and only if the diffeo A is semi-iterable in the sense of having half its analytic invariants C_{n}, namely those with negative n, identically zero. ^{37}

For details and complements, see [E3]. The above results demand some very careful analysis. Strangely enough, they are harder to prove that the forthcoming, more complete, and far more satisfactory, results about paralogarithmic synthesis.

The above resurgence is of a highly unusual type. In [E3] it is dubbed “synthesis resurgence” and contrasted with two other, far more common, types : “equational resurgence”^{38} and “co-equational resurgence”^{39}. Yet, remarkably enough, all three types give rise to the same sort of acting alien algebras ALIEN^{act} := ALIEN/ALIEN^{nil}, which is to say that in all three cases, the ideal ALIEN^{nil} of ALIEN which annihilates the relevant resurgence algebra ^{40} is generated by derivations of the form

c_{n_{1}+n_{2}} Δ Δ_{n_{1}+n_{2}} − (n_{1} − n_{2}) c_{n_{1}} c_{n_{1}} [Δ Δ_{n_{1}}, Δ Δ_{n_{2}}]

37at least in the Examples 1 and 2 under discussion here. Incidentally, the above construction relates more directly to Example 2, but can be easily transposed to Example 1.

38because there the resurgence-bearing variable is the (or one of the) variables(s) of the equation under investigation.

39because it is roughly dual to the preceding. It is also known as parametric resurgence because there the resurgence-bearing variable is, typically, a singular perturbation parameter.

40ie the algebra generated by the resurgent functions under consideration and all their alien derivatives, of all orders.

for a well-defined system of (usually transcendental) structure constants {c_{n}}.

4.7 The need for one free parameter at least.

Perhaps the main lesson from this excursion into the unilateral case is this : in order to describe the divergence-resurgence pattern of the series B_{∗} , B^{∗} as elements of ℂ[[s^{−1}]] we are compelled to adduce elements of ℂ[[s]]. In other words, what goes on at one pole s = ∞ (and z = ∞) of the Riemann sphere is inextricably linked to what goes on at the other pole s = 0 (and z = 0). ^{41} This “antipodal pairing” is going to be an outstanding feature of “canonical” or “sperical” synthesis.

But another lesson emerges, this time from the binary and sesquilateral cases : no one-to-one correspondance {𝔸_{ω}} ↔ Ob^{can} that holds for all systems {𝔸_{ω}} – and not just ‘small’ ones – can be achieved without introducing at least one free parameter capable of restoring ‘smallness’ where it is missing. Fortunately, as we shall see, one parameter is enough.

5 Methods for non-canonical Object Synthesis.

Having seen the limitations of canonical hyperlogarithmic synthesis and before turning to the completely satisfactory answer of canonical paralogarithmic synthesis, let us recall a few basic results about non-canonical synthesis, if only to obtain a characterisation of the admissible systems of invariants.

5.1 Main and earliest method.

We go back to Example 1. For any shift-like diffeo f, let f_{+}^{∗} (resp f_{−}^{∗}) denote the Borel-Laplace sum of the normaliser f^{∗} that is defined and regular in a U-shaped domain containing an upper (lower) half-plane ℑz > const (resp ℑz < const) and two vertical half-planes |ℜz| > const. For our present purpose, it is convenient to express the invariants {𝔸_{n}} in the guise of two 2πi-periodic functions π^{+} and π^{−} defined each in some vertical half-plane, respectively right and left :

41despite the absence, generally speaking, of an analytic continuation, from one pole to the next, of the functions involved.

π^{+}(z) = z − ∑_{n∈ℕ^{+}} A′_{n} e^{−nz} ; ℜz ≫ +1  (140)
π^{−}(z) = z − ∑_{n∈ℕ^{−}} A′_{n} e^{−nz} ; ℜz ≪ −1  (141)

and linking the normalisers f_{±}^{∗} on their common half-planar domains of definitions :

π^{+} ∘ f_{−}^{∗}(z) = f_{+}^{∗}(z)  for ℜz ≫ +1  (142)
π^{−} ∘ f_{+}^{∗}(z) = f_{−}^{∗}(z)  for ℜz ≪ −1  (143)

The Fourier coefficients A′_{n} of π_{±} are not exactly the constants A′_{n} occuring in the resurgence equations (72),(73), but rather those that we would get if we were to write down (72),(73) with the lateral operator Δ Δ_{ω}^{±} instead of Δ Δ_{ω} ^{42}.

Each conjugacy class of shift-like diffeos f is characterised by an analytic pair (π^{+}, π^{−}) and conversely to each analytic pair (π^{+}, π^{−}) there answers an analytic conjugacy class of shift-like diffeos f.

The direct part of the statement is very easy, from (142),(143). As for the converse statement, the original proof ([E2],pp 450-456) splits into an infinitesimal and a global step.

The infinitesimal step starts from an arbitrary shift-like diffeo f with analytic invariants (π^{+}, π^{−}) as in (140),(141). Then, given any infinitesimal analytic perturbation (π^{+} + δπ^{+}, π^{−} + δπ^{−}), one proceeds to construct a corresponding infinitesimal perturbation f +δf via the normalisers f_{±}^{∗} +δf_{±}^{∗}. The construction goes like this :

42Up to a 2πi factor, these A′_{n} are “polar residues” (in the Borel plane) at the points n as approached from below or from above.

2πi \frac{δf_{+}^{⋆}(z)}{∂f_{+}^{⋆}(z)} = − ∫_{t^{+}}^{+∞} \frac{(\frac{δ^{+}π}{∂^{+}π}) ∘ f_{+}^{⋆}(y)}{∂f_{+}^{⋆}(y)} \frac{dy}{y − z}

+ ∫_{t^{−}}^{−∞} \frac{(\frac{δ π^{−}}{∂ π^{−}} ) ∘ f_{+}^{⋆}(y)}{∂f_{+}^{⋆}(y)} \frac{dy}{y − z}  (144)

2πi \frac{δf_{−}^{⋆}(z)}{∂f_{−}^{⋆}(z)} = − ∫_{t^{−}}^{+∞} \frac{(\frac{δ^{−}π}{∂^{−}π}) ∘ f_{−}^{⋆}(y)}{∂f_{−}^{⋆}(y)} \frac{dy}{y − z}

+ ∫_{t^{+}}^{−∞} \frac{(\frac{δ π^{+}}{∂ π^{+}} ) ∘ f_{−}^{⋆}(y)}{∂f_{−}^{⋆}(y)} \frac{dy}{y − z}  (145)

with ^{±}π standing for the reciprocal functions of π^{±} and with ±ℜ(t^{±}) ≫ 1. The construction clearly implies :

\frac{δ f_{+}^{⋆}}{∂ f_{+}^{⋆}} − \frac{δ f_{−}^{⋆}}{∂ f_{−}^{⋆}} = +\frac{(\frac{δ π^{+}}{∂ π^{+}} ) ∘ f_{−}^{⋆}}{∂ f_{−}^{⋆}} for ℜ(z) ≫ +1

\frac{δ f_{+}^{⋆}}{∂ f_{+}^{⋆}} − \frac{δ f_{−}^{⋆}}{∂ f_{−}^{⋆}} = −\frac{(\frac{δ π^{−}}{∂ π^{−}} ) ∘ f_{+}^{⋆}}{∂ f_{+}^{⋆}} for ℜ(z) ≪ −1

which means that f + δf and f_{±}^{∗} + δf_{±}^{∗}. verify the perturbed form of the normaliser’s characteristic equation (142),(143).

The global step consists in joining some elementary but non-trivial pair {π_{0}^{−}, π_{0}^{+}} for which f_{0} is known^{43} to a given pair {π_{1}^{−}, π_{1}^{+}} for which f_{1} is sought , by an λ-analytic path {π_{λ}^{−}, π_{λ}^{+}, f_{λ} , f_{λ,±}^{∗} , t_{λ}^{±}}_{λ∈[0,1]}. This is done by turning (144),(145) into a differential system in λ, with the integration end points t_{λ}^{±} suitably chosen, ie far enough to the right or left for the compositions π_{λ}^{±} ∘ f_{λ,∓}^{∗} and ^{±}π_{λ} ∘ f_{λ,±}^{∗} to make sense. For details, see [E2] ^{44}.

5.2 Malgrange’s method.

Using polar coordinates x = e^{i θ} r on ℂ with (θ, r) ∈ (𝕋, ℝ^{+}) and considering the blow-up 𝕊 := (𝕋, 0), Malgrange defines sheaves Γ_{p} on 𝕊 by letting Γ_{p,θ_{0}} contain all functions that are holomorphic in some small sectorial neigbourhood of 0 of bisectrix θ_{0} with an asymptotic expansion of the form x + ∑_{n≥p+1} a_{n} x^{n}. Then he sets Γ_{∞} := ∩ Γ_{p} and shows that solving (142),(143) reduces to a special case of the following statement : the image of the natural mapping H^{1}(𝕊, Γ_{∞}) → H^{1}(𝕊, Γ_{p}) reduces to the trivial element O_{H^{1}(𝕊,Γ_{p})}. Malgrange then derives that statement, without calculations, from the classical Newlander-Nirenberg theorem.

43if {π_{1}^{−}, π_{1}^{+}} corresponds to the unilateral (resp sesqui- or bilateral) case, then the obvious choice for {π_{0}^{−}, π_{0}^{+}} is to take an instance of unary (resp binary) hyperlogarithmic synthesis, as in §4.5.1 (resp §4.5.2).

44in [E] we took f^{nor}(z) = z + 1 instead of f^{nor}(z) = z + 2 πi here, which explains the slight differences in the two presentations.

5.3 The quasiconformal method.

Another early proof of the (non-canonical) synthesis theorem for identitytangent, one-dimensional diffeos was given by S. M. Voronin. Though independent of Malgrangre’s proof, it resembles it in that it solves the problem first in a smooth setting (C^{∞} with M., C^{1} with V.), then in a complex-analytic one. But the main tool here is the beautiful Ahlfors-Bers theory (see [A]) of quasiconformal mappings.^{45} Starting from any given pair {π^{+}, π^{−}}, Voronin constructs {f_{+}^{∗}, f_{−}^{∗}, f}, with all the required relations, on an abstract Riemann surface S. Then he shows that S is quasiconformally, and therefore conformally, equivalent to a punctured neighbourhood of 0 in ℂ. These quasiconformal methods were later extensively used in holomophic dynamics by authors like Douady, Hubbard, Lavaurs, etc.

5.4 Comparison.

Anteriority aside, the method outlined in §5.1 would seem to be more explicit, elementary, and direct. Above all, it has the merit of extending with very little modification not only to most other problems of analytic synthesis (like Malgrange and Voronin’s methods), but also to non-analytic ones (unlike Malgrange’s or Voronin’s methods, which are too dependent on “geometry”, ie on the multiplicative plane, to tackle situation when geometry disappears).

Consider for example shift-like diffeos, but formal ones, of Gevrey rather than analytic type. In E2 it was shown that for all Gevrey classes G_{t} between G_{0} (analytic) and G_{1}, there exist non-trivial Gevrey moduli (ie Gevrey conjugacy classes), with invariants 𝔸_{n} := e^{−n z} A_{n} ∂_{z} which, relative to any well-behaved basis of ALIEN, are characterised by growth bounds

|A_{n}| < C_{0} exp (C_{1} n^{\frac{1}{1−t}}) (0 < C_{i} < ∞)

45ie mappings f : Ω ⊂ ℂ → ℂ with locally integrable, first-order derivatives such that |∂z̄ f| ≤ k |∂_{z} f| almost everywhere for some k < 1.

Here, the geometric model no longer exists and geometric methods become useless, whereas in the convolutive model, things hardly change. Indeed, the integrals (144),(145) can easily be rephrased in terms of their Borel transforms, and we find ourselves dealing with normalisers f̂^{∗}(ζ) and δf̂^{∗}(ζ) which have exactly the same singularities and verify the same resurgence equations as in the analytic case, but differ only by their growth pattern at ∞ (away from the singularities) :

|f̂^{∗}(ζ)| < C_{0} exp (C_{1} |ζ|^{\frac{1}{1−t}}) (0 < C_{i} < ∞)

6 Four closely related challenges.

We shall discuss in this section four closely related challenges, pertaining to four different notions central to resummation theory – uniformising averages, alien derivations, resurgenge monomials, ramified growth – and arising from the need to simultaneously satisfy a series of conditions : C1: compatibility with the multiplicative structure C2: respecting realness C3: respecting lateral (exponential) growth C4: scale invariance which are much the same in all four cases, yet call for slight reformulations. For uniformising averages these conditions shall become A1, …, A4. For alien derivations : D1, …, D4. For resurgence monomials : M1, …, M4. For ramified growth : R1, …, R4.

Each demand is easy enough to meet in isolation, but combining them is more difficult, for reasons we shall try to make clear in a moment. Fortunately, there are ways of reconciling these demands, and we then speak of well-behaved averages, derivations, monomials, etc.

But let us first devote a section to the central difficulty in all these problem : overcoming faster-than-lateral growth.

6.1 The main obstacle: faster-than-lateral growth.

As pointed out, summation or accelero-summation yields at every step i a function φ̂_{i}(ζ_{i}) which has precisely the right rate of growth, ie the one that makes the next acceleration (or Laplace) possible. But this applies only to singularity-free axes or, on singularity-carrying axes, to the two lateral determinations.^{46} Most other determinations of φ̂_{i}(ζ_{i}), especially the ones that correspond to oft-crossing paths, tend to display slightly faster-than-lateral growth.

For instance, if the lateral growth of φ̂(ζ) is exponential (this is the growth required for Laplace integration), the other determinations generally admit no better uniform bounds than |φ̂(ζ)| ≤ γ_{0} e^{γ_{1}|ζ|| log ζ|} (so that they cannot be subjected to Laplace). Therefore, unless we resort to carefully honed averages m, the averaged function m φ̂(ζ) itself is going to display this slightly super-exponential growth.

This nuisance of faster-than-lateral growth is extremely common, generic almost. In the Dulac problem for instance, it affects nearly all transit maps G_{i} attached to summits of semi-hyperbolic type in a given polycycle.

In order to show just how prevalent and inescapable the phenomenon of faster-than-lateral growth is, let us adduce the simplest conceivable illustration, linked to as harmless an operation as inversion :

Ã(z) := ∑_{n≥1} n! z^{−n}  (146)
Â(ζ) := ∑_{n≥0} ζ^{n} = 1/(1 − ζ)  (147)
B̃(z) := ∑_{n≥1} A^{n}(z) = A(z)/(1 − A(z))  (148)
B̂(ζ) := ∑_{n≥1} Â^{⋆n}(ζ)  (149)

The divergent series Ã(z) verifies the Euler equation (1 + ∂_{z})Ã(z) = z^{−1} and its Borel transform Â(ζ), with its single pole at ζ = 1, is the simplest instance of a non-trivial resurgent function. Yet a simple Möbius transform turns Ã(z) into a series B̃(z), also solution of a first-order differential equation, but with a Borel transform B̂(ζ) that has singularities at every point ζ = n ∈ ℕ, with simple poles as leading terms and logarithmic singularities as corrections :

B̂(ζ) = +R_{B}^{ϵ_{1},...,ϵ_{n−1},⋆}.(ζ − n)^{−1}  (simple pole)

+Reg_{1}^{ϵ_{1},...,ϵ_{n−1},⋆}(ζ − n). log(ζ − n) (logarithmic singularity)

+Reg_{0}^{ϵ_{1},...,ϵ_{n−1},⋆}(ζ − n) (regular part)

46and that too even if we take care of choosing in the critical time class {z_{i}} a suitably slow time z_{i} , which precaution has the effect of smoothing the singularities of φ̂_{i}(ζ_{i}).

[figure: A^{⋆}; A^{−⋆}; A^{+⋆}; A^{−−⋆}; A^{−+⋆}; A^{+−⋆}; A^{++⋆}]

This even provides us with a discretised model of the phenomenon of fasterthan-lateral growth. Indeed, the residues R_{B}^{ϵ_{1},...,ϵ_{n−1},⋆} of address {ϵ_{i}} are calculable by a simple induction (see [E8]) which readily shows that they verify no better bounds than

|R_{B}^{ϵ_{1},...,ϵ_{n−1},⋆}| ≤ c_{0} c_{1}^{n} n!  (150)

Therefore, although B̂ has exponential growth (at most) on each singularityfree axis arg ζ = θ ≠ 0, on paths that criss-cross ^{47} the axis arg ζ = 0 (especially for constantly alternating ϵ_{i}’s) it admits no better bounds than |B̂(ζ)| ≤ γ_{0} e^{γ_{1}|ζ|| log ζ|}.

In [E8], §A4, pp195-197, details may be found about the asymptotics of the “load” ld(ϵ_{1}, …, ϵ_{r−1}, ∗), which up to a multiplicative constant is the same as that of the residue R_{B}^{ϵ_{1},...,ϵ_{n−1},⋆}. At a distance r from the origin, the larger values occur on “oft-crossing paths”, ie for sequences (ϵ_{1}, …, ϵ_{n−1}, ⋆) with frequent sign changes, and the largest value occur on ever-crossing paths, ie for strictly alternating sequences (±, ∓, ±, ∓ …, ∗). The corresponding loads are roughly equal to (π/8)^{r} r! and that is also the order of magnitude for values |B̂(ζ)| on the branches with the same ϵ-address .

6.2 Challenge 1 : Searching for well-behaved averages.

In the process of mono- or polycritical resummation, it is not the – usually highly ramified – Borel transform φ̂(ζ) as such that gets subjected to the Laplace or acceleration transform, but some suitable uniformisation m φ̂(ζ) of φ̂(ζ), evaluated on the relevant integration axis {arg z = θ_{0}}, which for simplicity we shall take to be ℝ^{+} (ie θ_{0} = 0).^{48}

[figure: ω_{1}; ω_{2}; ω_{3}; uniform function m φ̂(ζ)]
↑  ↑  ↑

47while avoiding small disks |ζ − n| < ρ < 1/2 of fixed radius around each singular points. But we may remove this restriction by reasoning on the function ∫∫ B̂(ζ) which remains continous at each n.

48this applies in particular to the theory of transseries and analysable functions, where integration always takes place on ℝ^{+}.

[figure: −; +; −−; −+; +−; ++; multiform function φ̂(ζ)]

A uniformising average m : φ̂ ↦ m φ̂ is defined by a system of weights m^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} subject to the self-consistency relations :

∑_{ϵ_{i} ∈\{+,−\}} m^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{i} \atop ω_{i}} {,..., \atop ,...,} {ϵr \atop ωr})} = m^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{i+1} \atop ω_{i}+ω_{i+1}} {,..., \atop ,...,} {ϵr \atop ωr})}  ∀i < r (151)
∑_{ϵ_{r} ∈\{+,−\}} m^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{i} \atop ω_{i}} {,..., \atop ,...,} {ϵr \atop ωr})} = m^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{i} \atop ω_{i}} {,..., \atop ,...,} {ϵ_{r−1} \atop ω_{r−1}} )}  (152)

and its action is as follows :

m φ̂(ζ) := ∑_{ϵ_{i}=±} m^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} φ̂^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})}(ζ)  (153)

with φ̂^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} denoting the determination of φ̂ on the branch of address (ϵ_{1}, …, ϵ_{r}) over the interval ζ ∈ ]ω_{1} + … ω_{r} , ω_{1} + … ω_{r+1}[ between two consecutive singularities ^{49}

To be really useful in the present context of Borel-Laplace resummation, or in the more general one of accelero-summation, a uniformising average must fulfill four main conditions :

A1 : It must respect convolution^{50} , ie m (φ̂_{1} ⋆ φ̂_{2}) = (m φ̂_{1}) ⋆ (m φ̂_{2})

A2 : It must respect realness, ie m φ̂(ζ), as a global function, must be real whenever φ̂(ζ), as a germ at +0, is real.

A3 : It must respect lateral growth, that is to say m φ̂(ζ) must not grow significantly faster than the two lateral determinations (right or left) of φ̂(ζ) along the positive real axis.

A4 : It should be scale invariant, ie commute with real dilatations ζ ↦ lζ

Such averages will be declared “well-behaved”.

49Of course, when the singularities of the minor φ̂ are not of integrable type, we have to supplement the present formula with an anologous one for the major φ̌.

50observe that we are dealing here with two slightly different interpretations of the convolution product : in m (φ̂_{1} ⋆ φ̂_{2}) we convolute two function germs near the origin, then use analytic (or cohesive) forward continuation to get a global ramified function, and lastly we uniformise it by means of m , whereas in (m φ̂_{1}) ⋆ (m φ̂_{2}) we directly convolute two global, uniform functions.

A1 is essential to get an algebra morphism^{51}. A2 is natural and, in many instances^{52}, indispensible. A3 ensures the convergence of the acceleration (or Laplace) integrals. A4 is less essential, but very natural.

A2’s translation in terms of weights is straightforward : the weights should change into their complex conjugates when all signs ϵ_{i} are changed. If real, the weights should remain unchanged. The same with A4 : keeping all signs unchanged while multiplying all gaps ω_{i} by the same positive factor l should leave the weights unchanged. As for A1 and A3 we shall see in a moment what they imply in terms of weights.

But first we must introduce convenient, non-redundant ways of defining averages. To any uniformising average m we attach the moulds^{53} :

rem^{ω_{1},...,ω_{r}} := (−1)^{r} m^{( {+ \atop ω_{1}} {,..., \atop ,...,} {+ \atop ωr})}  ( “right-lateral mould” ) (154)
lem^{ω_{1},...,ω_{r}} := (−1)^{r} m^{( {− \atop ω_{1}} {,..., \atop ,...,} {− \atop ωr})}  ( “left-lateral mould” ) (155)
nam_{ω_{∗},t_{∗}}^{t_{1},...,t_{r}} := ϵ_{1} … ϵ_{r} m^{( {ϵ_{1} \atop ω∗} {,..., \atop ,...,} {ϵr \atop ω∗})}  ( “neutral mould” ) (156)
with ϵ_{i} := sign(t_{i} − t_{i−1}) (∀i < r) and ϵ_{r} := sign(t_{r} − t_{∗})

Due to the self-consistency relations, both the right- and left-lateral moulds encapsulate all the information about the entire weight system \{m^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})}\}, and each one can be deduced from the other in a simple manner. As for the “neutral mould”, its upper indices t_{i} are real numbers, and so too are its lower index t_{∗}, while the ‘common gap’ ω_{∗} must of course be positive.

The initial consonants r/l/n in the names of our moulds stand of course for right/left/neutral ; the inner vowels e/a stand respectively for symmetrel/al (for that’s what our moulds will have to be) ; and the final m stands for mean value.

We are now in a position to state the criteria for well-behavedness.

The four following conditions are equivalent: A1 : the uniformising average m respects convolution A1_{1} : the right-lateral mould rem^{•} is symmetrel ^{54} A1_{2} : the left-lateral mould lem^{•} is symmetrel A1_{3} : the neutral mould nam_{ω_{∗},t_{∗}}^{•} is symmetral for all ω_{∗}, t_{∗}

51Operating exclusively with algebra morphisms is a must in all non-linear problems.

Indeed, if φ̃ be the formal solution of a non-linear differential equation E, then its sum φ under Borel-Laplace (or, in polycritical instances, under accelero-summation) is going to verify the original equation E if and only if every single step in the resummation process is an algebra morphism.

52eg physics, real geometry, analysability theory, the Dulac problem, etc.

53see §2.1

The two following conditions are equivalent: A2 : the uniformising average m respects realness A2_{1} : the moulds rem^{•} and lem^{•} are complex conjugate

The seven following conditions are equivalent: A3 : the uniformising average m respects lateral growth A3_{1} : we have universal bounds | rem^{ω^{≺ ≺}} | ≤ C_{1} (D_{1})^{r} A3_{2} : we have universal bounds | rem^{ω^{≻ ≻}} | ≤ C_{2} (D_{2})^{r} A3_{3} : we have universal bounds | lem^{ω^{≺ ≺}} | ≤ C_{3} (D_{3})^{r} A3_{4} : we have universal bounds | lem^{ω^{≻ ≻}} | ≤ C_{4} (D_{4})^{r} A3_{5} : we have universal bounds | nam_{ω_{∗},t_{∗}}^{t^{≺}}| ≤ C_{5} (D_{5})^{ω_{∗} r} A3_{6} : we have universal bounds | nam_{ω_{∗},t_{∗}}^{t^{≻}}| ≤ C_{6} (D_{6})^{ω_{∗} r} for some finite constants C_{i}, D_{i}.

Here, r always denotes the cardinal of the sequences involved. We recall that M^{•^{≺}} (resp M^{•^{≻}}) denotes the forward (resp backward) arborification of the mould M^{•}. In symmetric fashion, M^{•^{≺ ≺}} (resp M^{•^{≻ ≻}} ) denotes the forward (resp backward) contracting arborification of the mould M^{•}. The defining relations read :

M^{ω^{≺ ≺}} := ∑_{ω^{1}>> ω^{≺ ≺}} M^{ω^{1}}  ;  M^{ω^{≻ ≻}} := ∑_{ω^{2}>> ω^{≻ ≻}} M^{ω^{2}}  (157)

The above symbol ω^{≺ ≺} (resp ω^{≻ ≻} ) denotes any sequence {ω_{i}} with an arborescent (resp anti-arborescent) order on it, ie an order such that each element ω_{i} has at most one predecessor ω_{i_{−}} (resp one successor ω_{i_{+}} ), whereas the sums on the right-hand side extend to all totally ordered sequences ω^{1} (resp ω^{2}) that can be obtained from ω^{≺ ≺} (resp ω^{≻ ≻} ) with the possible contraction ω_{i}, …, ω_{j} ↦ ω_{i} + ⋯ + ω_{j} of several consecutive elements. For some details see §2.1 and for more details go to [E5],[EV2],[EV3].

The four following conditions are equivalent: A4 : the uniformising average m is scale-invariant. A4_{1} : the mould rem^{•} is a homogeneous function of ω. A4_{2} : the mould lem^{•} is a homogeneous function of ω. A4_{3} : the mould nam_{ω_{∗},t_{∗}}^{•} is independent of its first lower index ω_{∗}

Tempting answer: the standard (or uniform, or median) average.

54see §2.1

Let mur and mul^{55} be the right- and left-lateral averages, with weights :

mur^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := 1 if ϵ_{1} = ⋯ = ϵ_{r} = + (res 0 otherwise) (158)
mul^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := 1 if ϵ_{1} = ⋯ = ϵ_{r} = − (res 0 otherwise) (159)

Both mur and mul verify A1, A3 and even A4 ^{56}, but clearly not A4_{2} (preservation of realness). The half-sum 1/2 (mur + mul) does verify A2, but ceases to verify A1 : it is not a convolution average and so of no use at all, except in linear problems. However, since mur and mul connect under post-composition by alien automorphisms :

mul := mur . rul  with rul := 1 + ∑_{ω>0} Δ Δ_{ω}^{−}  (160)
mur := mul . lur  with lur := 1 + ∑_{ω>0} Δ Δ_{ω}^{+}  (161)

it is very tempting to restore the right-left symmetry without destroying multiplicativity, by setting :

mun := mul . (lur)^{\frac{1}{2}} ≡ mur . (rul)^{\frac{1}{2}}  (162)

A simple calculation shows that the new average mun has weights that do not depend on the gaps ω_{i}, but only on the total number (p, q) of (+, −) signs in the sequence ϵ: ^{57}

mun^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := \frac{Γ(p + \frac{1}{2})Γ(q + \frac{1}{2})}{Γ(r + 1)Γ(\frac{1}{2})Γ(\frac{1}{2})} = \frac{(2p)! (2q)!}{4^{p+q} p! q! (p + q)!}  (163)

This mun is known as median or standard or uniform average. Its very construction ensures at once that it verifies A1, A2 and A4. But it can be shown^{58} that it fails in regard to A3 . So, for certain applications at least (see §7.6), we will have to look for better averages.

6.3 Challenge 2 : Searching for well-behaved alien derivations.

The singularities of minors φ̂(ζ) in the Borel plane deserve close attention because :

55m for mean value , u for uniform , r/l/n for right/left/neutral.

56since mur and mul load only the lateral paths

57These properties, in fact, completely caracterise mun among all convolution and realness preserving averages.

58see §7.6

– they are (mainly) responsible for the divergence of φ̃(z) in the formal model – they command the asymptotic behaviour of φ(z) in the geometric model, along the borders of its regularity sectors

– they carry, in the shape of residue-like coefficients in front of their leading terms, the analytic invariants of φ ^{59}.

Hence the need for operators capable of measuring these singularities precisely and conveniently. Such operators do exist : they are the so-called alien derivations Δ_{ω}. These are determined by systems of weights d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{i} \atop ω_{i}} {,..., \atop ,...,} {ϵr \atop ωr})} subject to the self-consistency relations :

∑_{ϵ_{i} ∈\{+,−\}} d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{i} \atop ω_{i}} {,..., \atop ,...,} {ϵr \atop ωr})} = d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{i+1} \atop ω_{i}+ω_{i+1}} {,..., \atop ,...,} {ϵr \atop ωr})}  ∀i < r (164)
∑_{ϵ_{r} ∈\{+,−\}} d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{i} \atop ω_{i}} {,..., \atop ,...,} {ϵr \atop ωr})} = 0  (165)

and their action in the convolutive model is given by :

Δ_{ω} φ̂(ζ) := ∑_{ϵ_{i}=±} d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} φ̂^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})}(ζ + ω)  (166)

for ζ close to +0 and by analytic continuation in the large. There being no scope for confusion, we also use the same symbols to denote the alien derivations acting in the multiplicative models (formal or geometric), ie the pull-backs by Borel-Laplace of the operators Δ_{ω} as defined by (166).

To each system of alien derivations we may associate the moulds :

red^{ω_{1},...,ω_{r}} := (−1)^{r} d^{( {+ \atop ω_{1}} {,..., \atop ,...,} {+ \atop ωr})}  (“right-lateral mould”)(167)
led^{ω_{1},...,ω_{r}} := (−1)^{r} d^{( {− \atop ω_{1}} {,..., \atop ,...,} {− \atop ωr})}  (“left-lateral mould”) (168)
nad_{ω_{∗},t_{∗}}^{t_{1},...,t_{r}} := ϵ_{1} … ϵ_{r} d^{( {ϵ_{1} \atop ω∗} {,..., \atop ,...,} {ϵr \atop ω∗})}  ( “neutral mould” ) (169)
with ϵ_{i} := sign(t_{i} − t_{i−1}) (∀i < r) and ϵ_{r} := sign(t_{r} − t_{∗})

Due to the self-consistency relations, both the right- and left-lateral moulds encapsulate all the information about the entire weight system \{d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})}\}, and each one can be deduced from the other in a simple manner. As for the ‘neutral’ mould, due to the second self-consistency relation (165), it is actually independent of the second lower index t_{∗} which consequently we shall drop.

59or more precisely, of the equation or system of which φ happens to be the solution.

The initial consonants r/l/n in the names of our moulds stand of course for right/left/neutral ; the inner vowels e/a stand respectively for alternel/al (for that’s what our moulds will have to be) ; and the final d stands for derivation.

The four following conditions are equivalent: D1 : the alien operators Δ_{ω} in (166) are (first-order) alien derivations D1_{1} : the right-lateral mould red^{•} is alternel D1_{2} : the left-lateral mould led^{•} is alternel D1_{3} : the neutral mould nad_{ω_{∗}}^{•} is alternal for all ω_{∗}

The two following conditions are equivalent: D2 : the alien derivations Δ_{ω} respect realness D2_{1} : the moulds red^{•} and led^{•} are complex conjugate

The seven following conditions are equivalent: D3 : the system Δ_{ω} in (166) of (first-order) alien derivations is well-behaved D3_{1} : we have universal bounds | red^{ω^{≺ ≺}} | ≤ C_{1} (D_{1})^{r} D3_{2} : we have universal bounds | red^{ω^{≻ ≻}} | ≤ C_{2} (D_{2})^{r} D3_{3} : we have universal bounds | led^{ω^{≺ ≺}} | ≤ C_{3} (D_{3})^{r} D3_{4} : we have universal bounds | led^{ω^{≻ ≻}} | ≤ C_{4} (D_{4})^{r} D3_{5} : we have universal bounds | nad_{ω_{∗}}^{t^{≺}}| ≤ C_{5} (D_{5})^{ω_{∗} r} D3_{6} : we have universal bounds | nad_{ω_{∗}}^{t^{≻}}| ≤ C_{6} (D_{6})^{ω_{∗} r} with the ordinary/contracting arborification rules of §2.1 and positive constants C_{i}, D_{i}.

The four following conditions are equivalent: D4 : the system {Δ_{ω}} of alien derivations is scale-invariant. D4_{1} : the mould red^{•} is a homogeneous function of ω. D4_{2} : the mould led^{•} is a homogeneous function of ω. D4_{3} : the mould nad_{ω_{∗},t_{∗}}^{•} is independent of its first lower index ω_{∗}

Tempting answer: the standard alien derivations.

In view of the Leibniz relations (56) verified by the right- and left-lateral differential operators Δ Δ_{ω}^{∓}, their sums rul, lur are mutually inverse alien automorphisms :

rul . lur = 1 , rul := 1 + ∑_{ω>0} Δ Δ_{ω}^{−} , lur := 1 + ∑_{ω>0} Δ Δ_{ω}^{+}  (170)
rul(φ̂_{1} ⋆ φ̂_{2}) ≡ rul(φ̂_{1}) ∗ rul(φ̂_{2})  (171)
lur(φ̂_{1} ⋆ φ̂_{2}) ≡ lur(φ̂_{1}) ∗ lur(φ̂_{2})  (172)

whose common logarithm resolves into a sum of true (ie first-order) alien derivations Δ Δ_{ω} = dun_{ω}:

+log(lur) ≡ −log(rul) =: 2πi dun ≡ 2πi ∑_{ω>0} dun_{ω}  (173)

They are the so-called standard or uniform alien derivations. A simple calculation shows that their weights are indeed the ones given in (57). As already pointed out, they are independent of the gaps ω_{i} and assign symmetric roles to the ± signs^{60}. From the way they are constructed, the standard alien derivations clearly verify D1 , D2 , D4, but not D3.

Let us explain why. Return to Example 1 of §3.1. Write the resurgence equation (72) successively for

Δ Δ_{n} := mun_{n} , Δ Δ_{n}^{+} := lur_{n} , Δ Δ_{n}^{−} := rul_{n}

and denote by A_{n} , A_{n}^{+} , A_{n}^{−} the corresponding coefficients that appear on the right-hand side. Whereas, as we saw in §4, the ‘lateral’ coefficients 𝔸_{n}^{±} give rise to convergent Fourier mappings :

z ↦ z + ∑_{0<n} A_{n}^{±} e^{−n z} (ℜz ≧ 1) ; z ↦ z + ∑_{n<0} A_{n}^{±} e^{−n z} (ℜz ≦ −1)

the ‘median’ coefficients 𝔸_{n}^{±} give rise to formal series :

2πi ∑_{0<n} A_{n} e^{−n z} (ℜz ≧ 1) ; 2πi ∑_{n<0} A_{n} e^{−n z} (ℜz ≦ −1)

which are the infinitesimal generators of the former, and therefore generically divergent, with Gevrey-1 bounds |A_{n}| ≤ C_{0} e^{C_{1} n} n^{n} instead of the exponential bounds |A_{n}| ≤ C_{0} e^{C_{1} n} which well-behaved alien derivations ought to ensure.

6.4 Challenge 3 : Searching for well-behaved resurgence monomials.

In §2.4 we already pointed to the existence of two basic sorts of resurgence monomials, the ∂- and Δ-friendly sorts, and we produced some elementary examples. Here, we are specifically interested in the the notion of Δ-friendly and ‘well-behaved’ resurgence monomials. So let us recall, and sharpen, the main demands we are making on these systems {U^{ω}(z) = U^{ω_{1},...,ω_{r}}(z)} of ‘atom-like’ resurgent functions. We want our monomials :

(1) to be as elementary as possible, and free of unnecessary parameters (2) to behave simply under multiplication or convolution^{61} (3′) to behave simply under alien differentiation, and not too badly under natural differentiation (4′) to be “complete” in the sense of enabling us to expand (or approximate) any given resurgent function φ :

60These two properties even characterise the standard derivations

φ(z) “ = ” ∑_{ω}c_{ω}(z) U^{ω}(z) := ∑_{r≥0} ∑_{ω_{i}} c_{ω_{1},...,ω_{r}}(z) U^{ω_{1},...,ω_{r}}(z) (174)

with coefficients c_{ω}(z) that are either ordinary constants or “resurgence constants”, that is to say functions with only vanishing alien derivatives : Δ_{ω_{0}} c_{ω}(z) ≡ 0 , ∀ω_{0}.

Condition (2) means that :

U^{ω′} U^{ω′′} ≡ ∑_{ω∈ sha(ω′, ω′′)} U^{ω}  (175)

with a sum extending to all sequences ω obtained by shuffling the two factor sequences ω′, ω′′. In other words, the mould U^{•} should be symmetral.^{62}

Condition (3’) is relative to a given basis {Δ_{ω}} of the algebra ALIEN of alien derivations. In concrete terms this condition stipulates that :

Δ_{ω_{0}} U^{ω_{1},...,ω_{r}} ≡ U^{ω_{2},...,ω_{r}}  if ω_{0} = ω_{1}
≡ 0  if ω_{0} ≠ ω_{1}  (176)

Clearly, there exist infinitely many multiplicative systems of resurgence monomials. Indeed, if U^{•}(z) is one such system, so will be the system U_{C}^{•}(z) := U^{•}(z)×C^{•}(z) derived therefrom by postmultiplication^{63} by any symmetral, resurgence-constant mould C^{•}(z). Multiplicative systems of resurgence monomials are extremely useful for solving resurgence equations, or systems of such equations, and to express their solutions in the form of expansions of type (174), often with constant coefficients c_{ω}. Thus, if we revert to Example 1 and try to solve the system of resurgence equations (113) that characterise the normalising transformation f^{⋆}, we find :

f^{⋆}(z) := z − ∑_{r} ∑_{n_{i}} A_{n_{1}} … A_{n_{1}} Γ_{n_{1},...,n_{r}} U^{n_{1},...,n_{r}}(z) (177)
with Γ_{n_{1},...,n_{r}} := (n_{1}) (n_{1} + n_{2}) … (n_{1} + n_{2} + ⋯ + n_{r−1})  (178)

61depending on the model.

62see §6.1

63in the sense of moulds.

But the real issue of course is convergence. We might try to solve it on an ad hoc basis, ie by choosing our resurgence monomials differently for each problem. But we are more ambitious : we ask for resurgence monomials that work in all cases. ^{64} That may seem a tall order, but it is feasibble ! The answer lies in the notion of well-behaved systems of resurgence monomials. And not only do such systems exist, but there is a canonical choice !

To any given system of resurgence monomials we may associate a rightlateral mould ^{re}U^{•}(z) and a left-lateral mould ^{le}U^{•}(z) characterised by the orthogonality conditions :

{^{re}U^{ω_{1},...,ω_{r}}} orthogonal to {Δ_{ω_{1},...,ω_{r}}^{+} := Δ_{ω_{r}}^{+} … Δ_{ω_{1}}^{+}}  (179)
{^{le}U^{ω_{1},...,ω_{r}}} orthogonal to {Δ_{ω_{1},...,ω_{r}}^{−} := Δ_{ω_{r}}^{−} … Δ_{ω_{1}}^{−}}  (180)

The three following conditions are equivalent: M1 : the system of resurgence monomials is multiplicative M1_{1} : the right-lateral mould ^{re}U^{•} is symmetrel^{65} M1_{2} : the left-lateral mould ^{le}U^{•} is symmetrel

Observe that the criterion here is symmetrel and not symmetral as in §3.6.1 This is because the lateral alien operators Δ_{ω}^{±} are not first-order alien derivations (see §3.5.1) and verify instead the Leibniz rules (56).

The two following conditions are equivalent: M2 : the system of multiplicative resurgence monomials is real M2_{1} : the moulds ^{re}U^{ω} and ^{le}U^{ω} are complex conjugate

The five following conditions are equivalent: M3 : the system of multiplicative resurgence monomials is well-behaved M3_{1} : we have universal bounds ∥^{re}U^{ω^{≺ ≺}} ∥ ≤ C_{1} (D_{1})^{r(ω^{≺ ≺} )} M3_{2} : we have universal bounds ∥^{re}U^{ω^{≻ ≻}} ∥ ≤ C_{2} (D_{2})^{r(ω^{≻ ≻} )} M3_{3} : we have universal bounds ∥^{le}U^{ω^{≺ ≺}} ∥ ≤ C_{3} (D_{3})^{r(ω^{≺ ≺} )} M3_{4} : we have universal bounds ∥^{le}U^{ω^{≻ ≻}} ∥ ≤ C_{4} (D_{4})^{r(ω^{≻ ≻} )} for some suitable norm ∥.∥ and with contracting arborification.

The three following conditions are equivalent: M4 : the system of multiplicative resurgence monomials is scale invariant M4_{1} : U^{ω_{1},...,ω_{r}}(z) ≡ U^{l ω_{1},...,l ω_{r}}(l^{−1}z) (∀l; in the multiplicative models) M4_{2} : Û^{ω_{1},...,ω_{r}}(z) ≡ l . Û^{l ω_{1},...,l ω_{r}}(l ζ) (∀l; in the convolutive model)

64ie for all systems of resurgence equations that do admit solutions.

65see §6.1.

Tempting answer: the standard resurgence monomials.

The standard or hyperlogarithmic monomials introduced in §2.4 (see also §4.1) cannot be beaten for simplicity. They also verify M1, M2, M4. But, as we saw in §4 when attempting to harness them for canonical synthesis, they don’t verify M3. So they are not the real answer.

6.5 Challenge 4 : Searching for the proper notion of ramified exponential growth.

Conditions R2 and R4 (respecting realness and scale invariance) are automatically fulfilled for all sensible growth conditions.

Conditions R1 means stability under the convolution product: if φ^{⋄}_{1} and φ^{⋄}_{2} possess minors that verify our growth condition, so too should φ^{⋄}_{1} ⋆ φ^{⋄}_{2}. This condition has two sides to it. In the absence of singularities, or in the presence of a finite number of them, or again of soft (eg integrable) singularities, the stability of the growth regimen at infinity is wholly unproblematic : it is automatically verified under very mild ‘convexity assumptions’ on the growth modulus. Difficulties arise only in the presence of fiercely explosive singularities (of a type which fortunately doesn’t occur in most applications, such as analysability theory, [E6],[E7]) and then only on paths that keep dangerously close to an infinite number of such singularities. This – largely academic – problem is examined in §9.3. It is essentially a question of controlling singularity explosiveness, or of keeping clear of singularities. But the “growth at infinity” aspect is wholly unproblematic : with the above reservations, naive exponential growth is stable under convolution.

So the crux is R3. It should be rephrased here as “stability under natural operations” such as inversion ( ie convolutive inversion in the Borel plane) or the solving of differential equations (ie their translations in the convolutive model) etc. But actually, for most growth conditions, including the one we shall settle for in §9, the seemingly weaker requirement of “stability under (convolutive) inversion” suffices to ensure the other ones. The main difficulty here has nothing to do with the explosiveness or density of the singularities in the Borel plane. It originates, rather, in the phenomenon of “faster-than-lateral” growth which was already pointed out in §6.1 and which comes into play practically each time we have a Borel axis arg ζ = θ that carries an infinity of singular points.

Tempting answer: straightforward exponential bounds.

The naive answer would be to define ramified-exponential growth as ordinary exponential growth along any broken line Γ starting at the origin, stearing clear of singularities ω in a uniform way (ie coming no closer to them than, say, ρ or ρ |ω| ), and with segments remaining within given angular bounds θ_{1} ≤ θ ≤ θ_{2}, with uniform bounds in θ_{1}, θ_{2}, ρ^{66}:

|φ̂(ζ)| ≤ C_{θ_{1},θ_{2},ρ} exp (D_{θ_{1},θ_{2},ρ} |ζ|)  (181)

or in the case of ramified-accelerable growth, to impose some analogous, but weaker bounds, like for instance :

|φ̂(ζ)| ≤ C_{θ_{1},θ_{2},ρ} exp (D_{θ_{1},θ_{2},ρ} |ζ|^{\frac{1}{1−α}})  (0 < α < 1)  (182)

for elementary accelerations z ↦ z^{\frac{1}{α}}

But all these definitions are unsuitable because, while meeting conditions R1, R2, R4, they completely fail with respect to R3 : see the introductory para §6.1.

6.6 Proofs.

The alien operators op in this section are all of the “exponential-bearing” sort, like Δ Δ_{ω}, Δ Δ_{ω}^{±}, etc. Their interpretation in the multiplicative models offers no difficulty, except that they act internally, not on the algebra RESUR of resurgent functions, but on RESUR tensored by exponential symbols :

RESUR ⊗ EXP := ⊕_{ω} RESUR_{ω} := ⊕_{ω} e^{−ω z} RESUR  (183)

In convolutive model these operators are called “stationary” because, unlike with the ordinary alien operators, their action there involves no translation : op φ̂(ζ) is always a finite sum of type ∑ γ_{i} φ̂(ζ_{i}) with points ζ_{i} that are all located over ζ. The exact interpretation in this model demands some care and a number of auxiliary constructions (see [E8]) which we won’t recall here, because we require only a few elementary facts about them.

The first fact is the existence, for each op , of a unique decomposition into a sum^{67} of ‘homogeneous’ components op_{ω} characterised by :

op = ∑_{0≤ω} op_{ω}  with op_{0} ∈ ℂ . id and  (184)
{ω_{0} > 0 , φ̂ regular over ω_{0}} ⇒ {op_{ω_{0}} φ ≡ 0}  (185)

66uniform bounds in θ_{1}, θ_{2} are indispensible even in the case of ordinary, non-ramified exponential growth.

67the sum may be infinite, but when applied to any given test function, it produces only finitely many non-zero terms.

The second fact is the possibility of characterising each op by a system of weights op^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} subject to the self-consistency relations (164),(165)^{68}.

The third fact is the existence of “complete” operators op^{∗} whose components op_{ω}^{∗} generate ALIEN , so that any other op can be expressed uniquely in terms of op^{∗} with the help of a “transit mould” ⟨op , op^{∗}⟩^{•} like this :

op = ∑ op_{ω} = ∑ ⟨op, op^{∗}⟩^{ω_{1},...,ω_{r}} op_{ω_{r}}^{∗} … op_{ω_{1}}^{∗}  (186)

The fourth fact is the existence of uniquely defined operators op = ({m_{1} \atop m_{2}} ) connecting by post-composition any pair m_{1}, m_{2} of uniformising averages :

m_{2} = m_{1} ({m_{1} \atop m_{2}} )  (187)

The fifth fact is the existence of the special, elementary operators :

lur := ({mul \atop mur}) = 1 + ∑_{0<ω} Δ Δ_{ω}^{+}  (188)
rul := ({mur \atop mul}) = 1 + ∑_{0<ω} Δ Δ_{ω}^{−}  (189)

which are automorphisms (see (56)) and complete.

The sixth fact is the simple interpretaion of the transit moulds < ., . >^{•} relative to lur or rul in terms of operator weights^{69}:

⟨op , rul⟩^{ω_{1},...,ω_{r}} = (−1)^{r} op^{( {+ \atop ω_{1}} {,..., \atop ,...,} {+ \atop ωr})}  (190)
⟨op , lur⟩^{ω_{1},...,ω_{r}} = (−1)^{r} op^{( {− \atop ω_{1}} {,..., \atop ,...,} {− \atop ωr})}  (191)

If we now recall the constrution of the right/left lateral moulds, we find for an alien derivation d these expressions :

red^{•} = ⟨d , rul⟩^{•}  (192)
led^{•} = ⟨d , lur⟩^{•}  (193)

68and not (151),(152). On the other hand, due to “stationarity”, the action of op in the convolutive model is not like (166) but more like (153), ie without ω shift.

69Heed the signs ! In (188),(189) lur , rul are associated with the signs +, − respectively, but in (190),(191) it is the reverse.

and for a uniformising average m we find these :

rem^{•} = ⟨\binom{mul}{m} , rul⟩^{•} = ⟨\binom{mul}{m} , \binom{mur}{mul}⟩^{•}  (194)
lem^{•} = ⟨\binom{mur}{m} , lur⟩^{•} = ⟨\binom{mur}{m} , \binom{mul}{mur}⟩^{•}  (195)

Criteria for C1 :

Using the interpretation of the four basic symmetry types for moulds (see §2.1.3 and below ; of course, we may replace field by derivation and diffeo by automorphism) and remembering that the alien operators lur, rul are automorphisms, we see that an alien operator d is a derivation if and only if the transit moulds (192),(193) are alternel; and that a uniformising average m respects convolution if and only if the transit moulds (194),(195) are symmetrel. This establishes the equivalent criteria given in §6 for A1 and D1. As for M1 and E1, the equivalent criteria are immediate.

Criteria for C2 and C4 :

All proofs are totally elementary here – the criteria are mere rephrasings.

Criteria for C3 :

Respecting “lateral growth”^{70} means that m φ̂(ζ) should not grow significantly faster than the right and left determinations mur φ̂(ζ) or mul φ̂(ζ) (ie exponentially in a monocritical problem). But in view of what precedes, the three averages are interrelated as follows :

m φ̂ (ζ) = ∑(rem^{ω_{1},...,ω_{r}}) (mul Δ Δ_{ω_{r}}^{−} … Δ Δ_{ω_{1}}^{−} φ̂ (ζ))  (196)
m φ̂ (ζ) = ∑(lem^{ω_{1},...,ω_{r}}) (mur Δ Δ_{ω_{r}}^{+} … Δ Δ_{ω_{1}}^{+} φ̂ (ζ))  (197)

or, in compact mould-comould form :

m φ̂ (ζ) = ∑(rem^{•}) (mul Δ Δ_{•}^{−} φ̂ (ζ))  (198)
m φ̂ (ζ) = ∑(lem^{•}) (mur Δ Δ_{•}^{+} φ̂ (ζ))  (199)

The trouble is that in practically all non-linear problems,the comould part mur//mul Δ Δ_{•}^{±} φ̂ (ζ) has faster-than-lateral growth and there is no way we can choose m to make rem^{•}//lem^{•} small enough to offset this effect^{71}. So instead of asking for the normal convergence of (198),(199) we should ask for that of the (formally equivalent) arborified expansions :

70in the Borel plane, along axis arg ζ = θ_{0}. Think of θ_{0} as 0.

71more precisely if – as we must here – we impose A1 and A2 on m : see §7.6.

m φ̂ (ζ) = ∑(rem^{•^{≺ ≺}} ) (mul Δ Δ_{•^{≺ ≺}}^{−} φ̂ (ζ))  (200)
m φ̂ (ζ) = ∑(lem^{•^{≺ ≺}} ) (mur Δ Δ_{•^{≺ ≺}}^{+} φ̂ (ζ))  (201)

Of course it has to be contracting arborification since rem^{•}//lem^{•} shall be chosen symmetrel (due to A1) and since Δ Δ_{•}^{−}//Δ Δ_{•}^{+} is co-symmetrel anyway.

The ultimate justification behind this choice is that it works. And it works because, in practically all natural situations, the resurgence is governed by some form or other of the Bridge Equation, which says that alien differentiation Δ Δ_{•} or Δ Δ_{•}^{±} amounts to ordinary differentiation 𝔸_{•} or 𝔸_{•}^{±} relative to the resurgence variable z and a finite number of parameters u_{i} ^{72}.

Now, it is a general fact that coarborification succeeds in drastically decreasing the norm of ordinary differential operators. And it is another fact that we can almost always manage to construct moulds – no matter what natural constraints they are subject to, like here symmetrelity plus A2 etc – in such a way as to prevent a norm increase under arborification.

This, roughly, is the motivation behind the decision to interpret the (vague) condition A3 in the form of the (precise and mutually equivalent) conditions A3_{1}, …, A3_{6} ^{73}.

The same applies for alien derivations and resurgence monomials. In the coming sections, we shall show how to construct uniformising averages, alien derivations and resurgence monomials in conformity with these conditions. Then, we shall tackle a particular application – Canonical Object Synthesis – and show in detail how arborification-coarborification works.

72ie those parameters that enter the complete, parameter-saturated solution of the problem. The operators 𝔸_{•} themselves depend on the problem, and so does the precise form of the Bridge Equation. In that deeper sense, and despite the ‘Bridge phenomenon’, alien calculus is totally irreducible to ordinary differential calculus, no matter what some adepts of Galois theory may claim.

73one point should be emphasised though : the conditions in the pairs A3_{1} and A3_{2}, or A3_{3} and A3_{4}, or A3_{5} and A3_{6}, would not be equivalent if they were to bear on general armoulds. But here, majorising the arborification or the antiarborification amounts to the same because the armoulds in question are induced by moulds. These moulds, moreover, are either symmetrel (like the lateral moulds) or symmetral (like the neutral mould), which further simplifies things, by essentially reducing the passage arborification → coarborification to multiplicative inversion.

7 Well-behaved uniformising averages.

7.1 Reminder about the standard or median average.

The standard or median average mun = mu_{\frac{1}{2} , \frac{1}{2}} can be embedded into a one-parameter family of so-called ‘uniform’ averages mu_{α,β} (with α + β = 1) that join the right-lateral average mur = mu_{1,0} and the left-lateral average mul = mu_{0,1} :

mu_{α,β}  := mur . (rul)^{β} ≡ mul . (lur)^{α}  (α + β = 1)
mu_{α,β}^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := \frac{Γ(p+α) Γ(q+β)}{Γ(r+1) Γ(α) Γ(β)}  (p + q = r)
mun^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := mu_{\frac{1}{2} , \frac{1}{2}}^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} = \frac{(2p)! (2q)!}{4^{p+q} p! q! (p+q)!}  (p + q = r)

We call these averages ‘uniform’ because their weights depend on the number (p, q) of (+, −) signs, but not on the gaps ω_{i}. Quite clearly, they always verify A1, A4, but they verify A2 only if α = β̄ (⇒ ℜ α = ℜ β = 1/2), and A3 only if α, β ∈ ℤ (see §7.5 , §8.7). These last two conditions being incompatible, there exists no well-behaved average in the uniform family.

7.2 Diffusion-induced averages.

Let {g_{ω}(•) , ω > 0} be some multiplicative semigroup and consider its Fourier transform, the convolution semigroup {f_{ω}(•) , ω > 0}:

g_{ω}(y) := e^{−ω γ(y)}  (y ∈ ℝ) (202)
f_{ω}(x) := \frac{1}{2π} ∫_{ℝ} g_{ω}(y) e^{ixy} dy  (x ∈ ℝ) (203)
g_{ω_{1}}(y) g_{ω_{2}}(y) ≡ g_{ω_{1}+ω_{2}}(y)  (204)
(f_{ω_{1}} ∗ f_{ω_{2}})(x) := ∫_{−∞}^{+∞} f_{ω_{1}}(x_{1}) f_{ω_{2}}(x − x_{1}) dx_{1} ≡ f_{ω_{1}+ω_{2}}(x) (205)

Let γ(y) be analytic on ℝ, vanish at y = 0 and have a fast increasing real part as y → ±∞ so that g_{ω}(.), f_{ω}(.) ∈ 𝕃^{1}(ℝ) with g_{ω}(0) = 1 and ∥f_{ω}∥_{𝕃^{1}} = 1. Let us view each function f_{ω}(.) as defining the ‘probability distribution’^{74} at the time t = ω, on the vertical axis ω + i ℝ, of a particle starting from the origin of ℂ at t = 0, moving along ℝ^{+} at uniform horizontal speed, and diffusing randomly in the vertical direction.

74strictly speaking, this applies only if f_{ω}(y) ≥ 0 but we can drop this restriction, because the numbers we shall proceed to define with the help of {f_{ω}(.)} neednot be real – they must simply verify certain algebraic relations (the ones implied by A1, A2, A4) and possess the right sort of bounds. That’s why we put ‘probability’ within inverted commas.

We may then define m^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr} )} as the probability of our particle’s successively crossing ω_{1} + i ℝ^{ϵ_{1}}, ω_{1} + ω_{2} + i ℝ^{ϵ_{2}} , …, ω_{1} + ω_{2} + … ω_{r} + i ℝ^{ϵ_{r}}.

[figure: ω_{1}; ω_{2}; ω_{3}; ω_{4}; ω_{5}; ω_{6}; 0; ϵ_{1}=+; ϵ_{2}=−; ϵ_{3}=+; ϵ_{4}=−; ϵ_{5}=−; ϵ_{6}=+]

Since these numbers m^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} verify the self-consistency relations (151) and (152), they may be regarded as weights defining a uniformising average m. That average clearly respects realness (condition A2) iff γ is an even function. Less obviously, it also respects convolution and lateral growth (conditions A1 and A3): see §7.5 infra.

7.3 Scale-invariance.

If we now select some τ > 0 and set γ(y) = ^{τ}γ(y) := |y|^{τ}, the variables x, y coalesce with the parameter ω:

τg_{ω}(y) ≡  ^{τ}g_{1}(ω^{\frac{1}{τ}} y) := exp(−ω |y|^{τ})  (y ∈ ℝ)(206)
τf_{ω}(x) ≡ ω^{− \frac{1}{τ}} ^{τ}f_{1}(ω^{− \frac{1}{τ}} x) := \frac{1}{2π} ∫_{ℝ} ^{τ}g_{ω}(y) e^{ixy} dy  (x ∈ ℝ)(207)

This automatically ensures the last missing condition A4 : invariance under a simultaneous, uniform dilation of all gaps ω_{i}.

As a consequence, for any τ ∈]0, +∞[ the corresponding uniformising average ^{τ}m is well-behaved.

Although this one-parameter family {^{τ}m} does contain all well-behaved, diffusion-induced aveages, there exist many well-behaved, averages which are not of this form, like for instance those of type :

m := ^{τ_{0}}m ({^{τ_{1}^{∗}}m \atop ^{τ_{1}}m})^{n_{1}} … ({^{τ_{k}^{∗}}m \atop ^{τ_{k}}m})^{n_{k}} (τ_{i}, τ_{i}^{∗} ∈ ℝ^{+} , n_{i} ∈ ℤ ; k ∈ ℕ) (208)

where ({• \atop •}) denotes the unique alien operator that connects any given pair of uniformising averages :

τ_{i}m := ^{τ_{i}^{∗}}m ({^{τ_{i}^{∗}}m \atop ^{τ_{i}}m})  (τ_{i}, τ_{i}^{∗} ∈ ℝ^{+})  (209)

But the most interesting average corresponds to the limit-case lim_{τ↓0} ^{τ}m.

7.4 The standard and organic averages as limit-cases.

For τ → +∞, ^{τ}m tends to the so-called “standard” average mun which as we saw is not well-behaved. For τ → +0 , ^{τ}m tends to the so-called “organic” average mon which as we shall see is well-behaved.

Though slightly more complex than the standard average, mon is by a long shot the simplest of all well-behaved averages. Its weights are given by the elementary recursion :

mo^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} = mo^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{r−1} \atop ω_{r−1}} )} P_{r}  (210)

with

P_{r} := 1 − \frac{1}{2} \frac{ω_{r}}{ω_{1} + … ω_{r}} if ϵ_{r−1} = ϵ_{r}  (211)
:= \frac{1}{2} \frac{ω_{r}}{ω_{1} + … ω_{r}}  if ϵ_{r−1} ≠ ϵ_{r}  (212)

Like mun, the organic mon can be imbedded into a one-parameter family of similar averages mo_{α,β} (α + β = 1) whose weights obey the following recursion :

mo_{α,β}^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := mo_{α,β}^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{r−1} \atop ω_{r−1}} )} \frac{ω_{1}+...ω_{r−1}+α ω_{r}}{ω_{1}+···+ω_{r}}  if (ϵ_{r−1}, ϵ_{r}) = (+, +)
:= mo_{α,β}^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{r−1} \atop ω_{r−1}} )}  \frac{β ω_{r}}{ω_{1}+···+ω_{r}}  if (ϵ_{r−1}, ϵ_{r}) = (+, −)
:= mo_{α,β}^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{r−1} \atop ω_{r−1}} )}  \frac{α ω_{r}}{ω_{1}+···+ω_{r}}  if (ϵ_{r−1}, ϵ_{r}) = (−, +)
:= mo_{α,β}^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{r−1} \atop ω_{r−1}} )} \frac{ω_{1}+...ω_{r−1}+β ω_{r}}{ω_{1}+···+ω_{r}}  if (ϵ_{r−1}, ϵ_{r}) = (−, −)
mon^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := mo_{1/2,1/2}^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} = 2^{−r} ∏_{i=2}^{i=r} (|ϵ_{i−1} + ϵ_{i}| − \frac{ϵ_{i−1} ϵ_{i} ω_{i}}{ω_{1} +···+ ω_{i}})

with |ϵ_{i−1} + ϵ_{i}| standing for 2 (resp. 0) if ϵ_{i−1} = ϵ_{i} (resp. ϵ_{i−1} ≠ ϵ_{i}).

Clearly mo_{1,0} = mu_{1,0} = mur and mo_{0,1} = mu_{0,1} = mul. Moreover :

All averages mo_{α,β} verify A1, A3, A4. In order for them to verify A2 and be well-behaved, we must have α = β̄ and so ℜ α = ℜ β = 1/2.

Let us add that F. Menous has devoted an entire PhD thesis ([Me]) to the subject of well-behaved uniformising averages. It contains in particular a meticulous investigation of the so-called Catalan average, which is not discussed here.

7.5 Proofs and comments.

Let us begin with the diffusion-induced averages. Applying the definition of §7.2 we get for the weights and the lateral moulds two types of integral expressions :

m^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := ∫_{ℝ^{r}} f_{ω_{1}}(x_{1}) … f_{ω_{r}}(x_{r}) σ_{ϵ_{1}}(x̌_{1}) … σ_{ϵ_{r}}(x̌_{r}) dx_{1} … dx_{r} (213)
:= \frac{ϵ_{1} … ϵ_{r}}{(2πi)^{r}} ∫_{ℝ_{ϵ_{1},...,ϵr}^{r}} \frac{g_{ω_{1}}(y_{1}) … g_{ω_{r}}(y_{r}) dy_{1} … dy_{r}}{(y_{1} − y_{2}) (y_{2} − y_{3}) … (y_{r−1} − y_{r}) y_{r}} (214)

rem^{ω_{1},...,ω_{r}} := ∫_{ℝ^{r}} f_{ω_{1}}(x_{1}) … f_{ω_{r}}(x_{r}) sofo x_{1},...,x_{r}_{+} dx_{1} … dx_{r}  (215)
:= \frac{(−1)^{r}}{(2πi)^{r}} ∫_{ℝ_{+,...,+}^{r}} g_{ω_{1}}(y_{1}) … g_{ω_{r}}(y_{r}) tas_{∞,0}^{y_{1},...,y_{r}} dy_{1} … dy_{r} (216)

lem^{ω_{1},...,ω_{r}} := ∫_{ℝ^{r}} f_{ω_{1}}(x_{1}) … f_{ω_{r}}(x_{r}) sofo x_{1},...,x_{r}_{−} dx_{1} … dx_{r}  (217)
:= \frac{1}{(2πi)^{r}} ∫_{ℝ_{−,...,−}^{r}} g_{ω_{1}}(y_{1}) … g_{ω_{r}}(y_{r}) tas_{∞,0}^{y_{1},...,y_{r}} dy_{1} … dy_{r} (218)

The sign functions σ_{±} are defined as in §2.1. The x-integrals are a direct rendition of the definition in §7.2, and always valid. The y-integrals are derived by Fourier transposition. They are easiest to interpret if g_{ω} is analytic on ℝ, for then y_{i} can be allowed to describe, in the course of integration, a minute half-circle around y_{i+1}, in the positive or negative direction depending on whether ϵ_{i} = + or −. When g_{ω} is merely C^{∞} on ℝ, this circumvention rule should of course be re-interpreted in terms of distributions.

Now let us turn to the uniform and organic averages. We leave the fact of their being limit-cases of ^{τ}m as a exercise, and choose instead to regard them as defined by their weights (210), which clearly satisfy the self-compatibility conditions. This leads for the lateral moulds to the following expressions :

remu_{α,β}^{ω} := (−1)^{r} \frac{Γ(r+α)}{(r+1)! Γ(α)}  ↔^{vol} lemu_{α,β}^{ω} := (−1)^{r} \frac{Γ(r+β)}{(r+1)! Γ(β)}
remun^{ω} := (−1)^{r} 4^{−r} \frac{(2 r)!}{r! r!}  ↔^{vol} lemun^{ω} := (−1)^{r} 4^{−r} \frac{(2 r)!}{r! r!}
remo_{α,β}^{ω} := ∏_{i=1}^{i=r}(\frac{β ω_{i}}{ω_{1}+···+ω_{i}} − 1) ↔^{vol} lemo_{α,β}^{ω} := ∏_{i=1}^{i=r}(\frac{α ω_{i}}{ω_{1}+···+ω_{i}} − 1)
remon^{ω} := ∏_{i=1}^{i=r}(\frac{\frac{1}{2} ω_{i}}{ω_{1}+···+ω_{i}} − 1) ↔^{vol} lemon^{ω} := ∏_{i=1}^{i=r}(\frac{\frac{1}{2} ω_{i}}{ω_{1}+···+ω_{i}} − 1)

We can now proceed to check the good-behaviour criteria.

Condition A1 is easiest to check in the form A1_{1} or A1_{2}. But the symmetrelity of rem^{•} or lem^{•} follows from the x-integral expressions (215),(217) and the symmetrelity of the moulds sofo•_{±}. One may also reason on the yintegrals (216),(218), noting that, due to the circumvention rules (of y_{i+1} by y_{i}), the symmetrality of the mould tas_{∞,0}^{•} translates into the symmetrelity of the integrals. This disposes of the diffusion-induced averages. For the limit cases mun and mon and their ‘tilted’ variants mu_{α,β} and mo_{α,β}, the symmetrelity of the lateral moulds can be checked directly.

Conditions A2 and A4 are immediate to check directly on the weights.

This leaves A3. Being a non-algebraic growth condition, it is the most difficult one. However, by resorting to the criteria A3_{2} or A3_{4}, it can be settled algebraically by observing that the mould sofo•_{±} retain their form under contracting anti-arborification : they simply become armoulds sofo•^{≻ ≻}_{±} that are still defined by the formula in §2.1.9, except that the sums x̌_{i} must now be extended to all x_{j} anterior to x_{i} under the anti-arborescent order^{75}. Checking this in a simple exercise : reason inductively on the number of branches and make repeated use of the identities :

σ_{±}(a) σ_{±}(a + b) + σ_{±}(b) σ_{±}(a + b) = σ_{±}(a) σ_{±}(b) + σ_{±}(a + b)  (219)

Remark : We have been using the right- and left-lateral moulds simply for the sake of symmetry, but they are essentially the same. Indeed, as soon as as condition A1 is fulfilled, they are related under the simple involution :

rem^{•} ↔ lem^{•} : lem^{•} ≡ (rem^{•} ∘ J^{•}) × J^{•}

75 x_{i} itself should be included in the sum.

with J^{ω_{1},...,ω_{r}} := (−1)^{r}. The elementary mould J^{•} is itself symmetrel and involutive in the sense that J^{•} ∘ J^{•} = 1^{•} + I^{•}.

7.6 When exactly must we take recourse to well-behaved averages ?

Oft-crossing paths.

Let φ̂ possess infinitely many singularities over the zero axis arg ζ = 0 or ℝ^{+} for short, for instance at all integral points n. As our example in §6.1 suggests, and as a careful examination of the non-linear examples in §3 would confirm, the growth of φ̂(ζ) is bad on oft-crossing paths, and worst on the ever-crossing path Γ^{+−+−...} that intersects every interval between two consecutive singularites. There φ̂(ζ) grows roughly like e^{C_{1} |ζ|} Γ(ζ). But it so happens that on these very paths some (not all) well-behaved averages, in particular the organic average mon, possess weights that decrease precisely like e^{C_{1} |ζ|}/Γ(ζ). Thus, on these ever-crossing paths, functions and weights are perfectly matched, their product having exponential growth at most. The same function-weight matching also holds, trivially, for the never-crossing ie lateral paths. This raises the question : don’t there exist ‘super well-behaved’ or ‘magic’ averages, such that not only |m φ̂(ζ)| grows exponentially, but also the sum ∑_{Γ} |m_{Γ} φ̂^{Γ}(ζ)| over all the branches Γ ?

Role of sign compensations.

This, however, is impossible, as was shown in [E8] : There exist no magic averages , capable of acting by “brute force” alone. Strangely, the trouble comes from paths that cross ℝ^{+} neither too often nor too rarely^{76}.

Anyway, the upshot is that some quite systematic branch-to-branch compensation must come into play. This compensation, however, which explains the success of well-behaved averages m, doesn’t come from the average’s weights (all of which may be real positive, as with mon) but from the functions φ̂(ζ) which, constrained as they are by the Bridge Equation, display a precise pattern of sign alternation on their different branches.

Frequency of faster-than-lateral growth but infrequency of nonmedianisable growth.

One should carefully distinguish between the quite frequent phenomenon of faster than lateral growth on the Borel axis of direction θ, which tends to occur (in non-linear problems) as soon as there is one^{77} active alien derivation Δ Δ_{ω_{1}} on that axis (ie arg ω_{1} = θ), and the far rarer need to take recourse to a well-behaved average in order to Laplace-sum (or accelerate) over that same axis, which need arises only if there are infinitely many active alien derivations Δ Δ_{ω_{i}} on the axis (ie arg ω_{i} = θ). In the first case we speak of medianisable^{78} faster-than-lateral growth; and in the second case, of nonmedianisable faster-than-lateral growth.

76see [E8] Prop A.5.5.

77or several, but finitely many.

Four instances of medianisable growth.

med1 : Convolution inverses. This is the function B̂ in §6.1. The axis direction is θ = 0 and the only active derivation is Δ Δ_{ω_{1}} = Δ Δ_{1}

med2 : Euler-like equations in the “negative direction”. This is Ex 2 of §3.2 with θ = π mod 2π and Δ Δ_{ω_{1}} = Δ Δ_{−1}.

med3 : Singular Riccati equations in “both directions”. This is again Ex 2 of §3.2, but with only three non-zero coefficients b_{−1}(z), b_{0}(z), b_{1}(z). Then we have medianisable faster-than-lateral growth in two directions : for θ = π mod 2π with Δ Δ_{ω_{1}} = Δ Δ_{−1} and for θ = 0 mod 2π with Δ Δ_{ω_{1}} = Δ Δ_{1}.

med4 : Resonant systems in the “negative directions”. This is Ex 3 of §3.3 with θ = arg(−λ_{j}) mod 2π and Δ Δ_{ω_{1}} = Δ Δ_{−λ_{j}}.

In all four cases, despite the presence of infinitely many singular points (in arithmetical progression) on the Borel axis of direction θ, the alien automorphism lur_{θ} which links the lateral averages acts, due to the Bridge equation, like the convergent differential operator on the far right of (220), and so its square root acts like the equally convergent differential operator on the far right of (221).

\binom{mur_{θ}}{mul_{θ}} = lur_{θ} = exp (2πi ∑_{arg ω=θ} Δ Δ_{ω}) ∼ exp (2πi 𝔸_{ω_{1}})  (220)

\binom{mul_{θ}}{mur_{θ}}^{\frac{1}{2}} = lurθ^{\frac{1}{2}} = exp (πi ∑_{arg ω=θ} Δ Δ_{ω}) ∼ exp ( πi 𝔸_{ω_{1}})  (221)

We may therefore^{79} in all these cases, apply the median or standard average mun_{θ}: its weights will automatically combine with the superexponentially large values of φ̂(ζ) encountered on the various branches in such a way as to produce (at most) an exponentially large mun_{θ} φ̂(ζ)

Four instances of non-medianisable growth.

78because in that case the median average mun does respect lateral growth.

79in view of the interpretation (160),(161) of the median average mun .

The same would still hold for a finite number of active derivations on our Borel axis, but not for an infinite number. The reason is this : take ordinary first-order differential operators 𝔸_{ω} and their formal sum 𝔸 = ∑_{arg ω=θ} 𝔸_{ω}.^{80} Then, if the sum 𝔸 has an infinite number of terms, the convergence of the formal automorphism exp(2πi 𝔸) in no way implies that of its infinitesimal generators 𝔸 or even that of its square root exp(πi 𝔸). This explains why, in all such cases, we will generically be facing non-medianisable faster-than-exponential growth.

We may of course use the lateral averages mur_{θ} or mur_{θ} if we don’t object to imaginary parts, but if the Borel axis is ℝ^{±} and if the context makes imaginary parts inacceptable (as in physics, or real geometry, or the theory of analysable functions), there there is no alternative to the well-behaved averages. Here are four such examples.

non-med1 : unitary iteration of unitary diffeomorphisms. This is Ex 1 of §3.1, but for “unitary” diffeos, whose inverse coincides with their complex conjugate : f ∘ f̄ = id. If we want to define sectorial-regular fractional iterates of f in right or left half-planes, then we must Laplace integrate on ℝ^{+} or ℝ^{−} with respect to a well-behaved average^{81}.

non-med2 : Euler-like equations on the “positive directions”.

This is once more Ex 2 of §2 but in the direction θ = 0. If we consider the particular formal solution ỹ(z, 0) there is no probblem, since ŷ(ζ, 0) has no singularities on ℝ^{+}. But if we ask for a real resummation of the full solution ỹ(z, u) ^{82} then we have to contend with an infinity of active alien derivations and must use a well-behaved average.^{83}

non-med3 : hyperbolic transit maps. Let ϖ be a real, local-analytic differential form formally conjugate to ϖ^{nor} := \frac{dx_{2}}{x_{2}} + (1 + ρ x_{1})\frac{dx_{1}}{x_{1}^{2}} . Its integral curves are hyperbolae that draw closer and closer to the coordinate axes (in the first and third quadrants). The local “transit map” g : x_{1} ↦ x_{2} which link the distance to the axes (on suitable transversals) of the incoming and outcoming hyperbolae branches, corresponds to a formal transseries that commingles powers and exponentials and generically displays non-medianisable faster-than-lateral growth. These local transit maps play a part in the resummation-theoretic proof of the Dulac conjecture (about the finiteness of limit cycles for polynomial planar vector fields). For a detailed discussion, see [E8],[EM]. Here, however, it should be noted that the well-behaved average is needed to go from the geometric germ g (which is given by ϖ ) to its formalisation g̃. In other words, the usual process (from formal to geometric) is reversed.

80for example, take the 𝔸_{n} as in (74) or (79)

81see [EM], §4

82in domains where |z^{−1}| and |u e^{z}| are both small.

83a detailed discussion may be found in [E6], §3.10.

non-med4 : Resonant systems on the “positive directions”. This is again Ex 3 of §3.3 but this time on a “positive axis” θ = arg λ_{j}. The picture is much the same as in non-med2 : we have non-medianisable faster-than-lateral growth for ŷ(ζ, u) though not for ŷ(ζ, 0) or indeed for any ‘partial’ solution that leaves out the parameter u_{j}.

8 Well-behaved alien derivations.

8.1 Reminder about the standard alien derivations.

In §2.3 we introduced, and then dismissed, the lateral alien operators Δ_{ω}^{±} which do indeed measure singularities over ω and also satisfy D3, D4, but infringe D1 (they are not first-order derivations) and D2.

Then from the Δ_{ω}^{±} we constructed (§6.3, (172)) the more satisfactory standard, or uniform, alien derivations Δ_{ω}. These verify D1, D2, D4, but not D3 : they are not well-behaved. But they have very simple weights :

d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := \frac{ϵ_{r}}{2πi} \frac{p! q!}{(p + q + 1)!} = \frac{ϵ_{r}}{2πi} \frac{p! q!}{r!} {p := #\{1 ≤ i < r; ϵ_{i} = +\} \atop q := #\{1 ≤ i < r; ϵ_{i} = −\}}

and they relate to the standard averages as follows^{84} :

∂_{α}mu_{α,β} ≡ 2πi mu_{α,β} dun with α + β = 1 , dun := ∑_{ω>0} Δ Δ_{ω} (222)

Weight-wise, and after due order reversal, this translates into :

∂_{α}mu_{α,β}^{•} ≡ 2πi dun^{•} × mu_{α,β}^{•}  ie :  (223)
∂_{α}mu_{α,β}^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} ≡ 2πi ∑_{1≤i≤r} dun^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵ_{i} \atop ω_{i}})} mu_{α,β}^{( {ϵ_{i+1} \atop ω_{i+1}} {,..., \atop ,...,} {ϵr \atop ωr})} (224)

8.2 Diffusion-induced alien derivations.

We revert to the ‘diffusion process’ of §7.2, but now we define the number 2πi ϵ_{r} d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} as the conditional probability that a particle starting from 0 and crossing the real axis at the point ω_{1}+… ω_{r} should successively cross the vertical half-axes ω_{1} + i ℝ^{ϵ_{1}}, ω_{1} + ω_{2} + i ℝ^{ϵ_{2}} , …, ω_{1} + ω_{2} + … ω_{r−1} + i ℝ^{ϵ_{r−1}}.

84here, α + β ≡ 1 and so the derivation ∂_{α} = −∂_{β} acts on both α and β.

[figure: ω_{1}; ω_{2}; ω_{3}; ω_{4}; ω_{5}; ω_{6}; 0; ϵ_{1}=+; ϵ_{2}=−; ϵ_{3}=+; ϵ_{4}=−; ϵ_{5}=−; ϵ_{6}=+]

Since these numbers d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} verify the self-consistency relations (164),(165), they may be looked upon as the weights of a well-defined alien operators d. That operator clearly respects realness (condition D2) iff γ is an even function. Less obviously, it is also a first-order alien derivation (condition D1) and it even respects lateral growth (condition D3): see §7.5 infra.

8.3 Scale-invariance.

If we now switch to {^{τ}f_{ω} , ^{τ}g_{ω}} for 0 < τ < +∞ as we did in §7.3, then the corresponding systems {^{τ}d_{ω}}, while retaining D1,D2,D3, acquire scaleinvariance (condition D3) : in other words, they are well-behaved systems of alien derivations. From these, many other well-behaved systems can be constructed : not only can we resort to the same trick (208) as with the uniformising averages^{85} but, unlike with the averages, we can also take advantage of the stability of well-behaved alien derivations under addition and the Lie bracket.^{86}

8.4 Standard and organic alien derivations as limitcases.

For τ → +∞, the system ^{τ}d = {^{τ}d_{ω}} tends to the system dun = {dun_{ω}} of so-called “standard” alien deivations, which as we saw is not well-behaved.

For τ → +0 , the system ^{τ}d = {^{τ}d_{ω}} tends to the system don = {don_{ω}} of so-called “organic” alien derivations, which as we shall see is well-behaved.

85with d,^{τ_{0}} d in place of m,^{τ_{0}} m , but without changing the left-side factors ({^{τ^{∗}_{i}}m \atop ^{τ_{i}}m})^{n_{i}}.

86thus, setting d′_{ω} := a ^{τ_{1}}d_{ω} + b ^{τ_{2}}d_{ω} and d′′_{ω} := ∑_{ω_{1}+ω_{2}=ω} c(ω_{1}/ω_{2}) [^{τ_{1}}d_{ω_{1}} , ^{τ_{2}}d_{ω_{2}}] , with a bounded c, we get two new well-behaved systems {d′_{ω}} and {d′′_{ω}}.

Like the organic average mon, the organic system of alien derivations don has weights that are simple^{87} rational functions of the gaps ω_{i} and it can be embedded into a one-parameter family don_{α,β} (with α + β = 1) of systems which are charcterised by :

∂_{α} mo_{α,β} = −∂_{β} mo_{α,β} = 2πi mo_{α,β} do_{α,β}  (225)

and which are well-behaved iff α = β̄. The “tilted” systems do_{1,0}, do_{0,1} are particularly simple, since they load (each) only r out of 2^{r} paths. Unfortunately, they are not well-behaved (they offend against D2) but their half-sum dom is well-behaved, like don, only much simpler. In fact, there exists no simpler choice of well-behaved derivations than dom. As for the weights of the organic family, simple calculations lead to the following formulas :

don^{( {ϵ_{1} \atop ω_{1}} {... \atop ...} {ϵr \atop ωr})} := do_{1/2,1/2}^{( {ϵ_{1} \atop ω_{1}} {... \atop ...} {ϵr \atop ωr})}  with do_{α,β}^{•} characterised by
∂_{α} mo_{α,β}^{•}  = 2πi do_{α,β}^{•} × mo_{α,β}^{•} and α + β ≡ 1 , ∂_{α} ≡ −∂_{β}
dom^{•}  := \frac{1}{2} (do_{1,0}^{•} + do_{0,1}^{•})
dom^{( {ϵ_{1} \atop ω_{1}} {... \atop ...} {ϵr \atop ωr})} := \frac{1}{2} \frac{ϵ_{r}}{2πi} \frac{ω_{p+1}}{ω_{1}+···+ω_{r}}  if (ϵ_{1}, …, ϵ_{r}) = ((+)^{p}, (−)^{q}, ϵ_{r})
:= \frac{1}{2} \frac{ϵ_{r}}{2πi} \frac{ω_{q+1}}{ω_{1}+···+ω_{r}}  if (ϵ_{1}, …, ϵ_{r}) = ((−)^{q}, (+)^{p}, ϵ_{r})
:= 0  otherwise

8.5 Proofs and comments.

We begin with the diffusion-induced derivations. Applying the definition of §8.2 we get for the weights and the lateral moulds two types of integral expressions :

d^{( {ϵ_{1} \atop ω_{1}} {,..., \atop ,...,} {ϵr \atop ωr})} := \frac{ϵ_{r}}{2πi} ∫_{ℝ^{r}} f_{ω_{1}}(x_{1}) … f_{ω_{r}}(x_{r}) σ_{ϵ_{1}}(x̌_{1}) … σ_{ϵ_{r−1}}(x̌_{r−1})δ(x̌_{r}) dx_{1} … dx_{r}
:= \frac{ϵ_{1} … ϵ_{r}}{(2πi)^{r+1}} ∫_{ℝ_{ϵ_{1},...,ϵr}^{r}} \frac{g_{ω_{1}}(y_{1}) … g_{ω_{r}}(y_{r}) dy_{1} … dy_{r}}{(y_{1} − y_{2})(y_{2} − y_{3}) … (y_{r−1} − y_{r})}  (226)

87but less simple than with mon, for they admit no closed expression as a product of r elementary factors.

red^{ω_{1},...,ω_{r}} := \frac{1}{2πi} ∫_{ℝ^{r}} f_{ω_{1}}(x_{1}) … f_{ω_{r}}(x_{r}) lefo x_{1},...,x_{r}_{+} dx_{1} … dx_{r}  (227)
:= \frac{(−1)^{r}}{(2πi)^{r+1}} ∫_{ℝ_{+,...,+}^{r}} g_{ω_{1}}(y_{1}) … g_{ω_{r}}(y_{r}) tas_{∗∗}^{y_{1},...,y_{r}} dy_{1} … dy_{r}(228)

led^{ω_{1},...,ω_{r}} := \frac{−1}{2πi} ∫_{ℝ^{r}} f_{ω_{1}}(x_{1}) … f_{ω_{r}}(x_{r}) lefo x_{1},...,x_{r}_{−} dx_{1} … dx_{r}  (229)
:= \frac{−1}{(2πi)^{r+1}} ∫_{ℝ_{−,...,−}^{r}} g_{ω_{1}}(y_{1}) … g_{ω_{r}}(y_{r}) tas_{∗∗}^{y_{1},...,y_{r}} dy_{1} … dy_{r}(230)

Now let us turn to the alien derivations of uniform or organic type and form their lateral moulds. We find :

2πi redun^{ω} := (−1)^{r} \frac{1}{r}  ↔^{vol} 2πi ledun^{ω} := (−1)^{r−1} \frac{1}{r}
2πi redon^{•} := dremon^{•} × vremon^{•} ↔^{vol} 2πi ledon^{•} := dlemon^{•} × vlemon^{•}
2πi redom^{ω} := (−1)^{r} \frac{1}{2} \frac{ω_{1}+ω_{r}}{ω_{1}+···+ω_{r}}  ↔^{vol} 2πi ledom^{ω} := (−1)^{r−1} \frac{1}{2} \frac{ω_{1}+ω_{r}}{ω_{1}+···+ω_{r}}

dremo_{α,β}^{•} := ∂_{α} remo_{α,β}^{•}
dlemo_{•} := ∂_{α} remo_{α,β}^{•} = − ∂_{β} lemo_{α,β}^{•}
vremo_{α,β}^{•} := (remo_{α,β}^{•})^{−1}  (mould inversion)
vlemo_{α,β}^{•} := (lemo_{α,β}^{•})^{−1}  (mould inversion)

dremo_{α,β}^{ω_{1},...,ω_{r}} := + (∑_{i=1}^{i=r} \frac{ω_{i}}{ω_{1}+···+ω_{i−1}+α ω_{i}}) remo_{α,β}^{ω_{1},...,ω_{r}}
dlemo_{α,β}^{ω_{1},...,ω_{r}} := − (∑_{i=1}^{i=r} \frac{ω_{i}}{ω_{1}+···+ω_{i−1}+β ω_{i}}) lemo_{α,β}^{ω_{1},...,ω_{r}}
vremo_{α,β}^{ω_{1},...,ω_{r}} := + α \frac{ω_{1}}{ω_{1}+···+ω_{r}} ∏_{i=2}^{i=r}(α \frac{ω_{i}}{ω_{1}+···+ω_{i}} − 1)
vlemo_{α,β}^{ω_{1},...,ω_{r}} := + β \frac{ω_{1}}{ω_{1}+···+ω_{r}} ∏_{i=2}^{i=r}(β \frac{ω_{i}}{ω_{1}+···+ω_{i}} − 1)

We can now proceed to check the good-behaviour criteria.

Condition D1 is easiest to check in the form D1_{1} or D1_{2}. But the alternelity of red^{•} or led^{•} follows from the x-integral expressions (227),(229) and the alternelity of the moulds lefo•_{±}. One may also reason on the y-integrals (228),(229) by heeding the circumvention rules already mentioned in §7.5 and by observing that they have the effect of turning the alternal mould tas_{∗∗}^{•} into alternel integrals ! In the limit cases dun and don, as well as with dom, it is best to check the alternelity directly on the above formulas.

Conditions D2 and D4 are immediate to check directly on the weights.

This leaves the “difficult” growth condition D3. Again, though non-algebraic , this condition, once rephrased as criterion D3_{2} or D3_{4}, can be settled algebraically, by observing that the mould lefo•_{±} retain their form under contracting anti-arborification : they simply become armoulds lefo•^{≻ ≻}_{±} that are still defined as in §2.1.9, except that the sums x̌_{i} must now be extended to all x_{j} anterior to x_{i} under the anti-arborescent order^{88}. In particular, lefoω^{≻ ≻}_{±} = 0 as soon as the anti-arborescent sequence ω^{≻ ≻} has more than one anti-root or maximal element : this is the so-called property of separativity for alternel armoulds. Checking these identities is a simple, but useful exercise.^{89}

Remark : We have been using the right- and left-lateral moulds simply for the sake of symmetry, by they are essentially the same. Indeed, as soon as as condition D1 is fulfilled, they are related under the simple involution :

red^{•} ↔ led^{•} : led^{•} ≡ red^{•} ∘ J^{•}

with J^{ω_{1},...,ω_{r}} := (−1)^{r} , ∀ω_{i}.

8.6 Tables of averages and derivations.

The following tables contrast the behaviour of averages and derivations from the organic (well-behaved) and uniform (non w.-b.) families. The latter satisfy all conditions Ci except C3 (“proper growth”). This is reflected in the fact that for strongly alternating sign sequences ϵ, the weights of the organic (resp uniform) operators tend to be small (resp not so small). The difference would be even more striking if it were possible to print the tables for larger lengths (We had to stop at r = 6).

To simplify, we set all gaps ω_{i} equal to 1. So we mention only the ± signs ϵ_{i}. For derivations, we drop the \frac{1}{2πi} factor. We also take advantage of the left-right symmetries to retain only the sign sequences ending with ϵ_{r} = + :

m^{ϵ̄_{1},...,ϵ̄_{r}} ≡ + m^{ϵ_{1},...,ϵ_{r}}  ;  d^{ϵ̄_{1},...,ϵ̄_{r}} ≡ − d^{ϵ_{1},...,ϵ_{r}}  (231)

88 x_{i} itself should be included in the sum.

89alternatively, one may reason on the y-integrals and observe first that the mould tas_{∗∗}^{•} retains its form under ordinary (ie non-contracting) anti-arborification and second that, under the circumvention rules, this translates into contracting anti-arborification for the integrals. But, on balance, reasoning on the x-integrals is simpler.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
object  average average derivation derivation derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
species uniform organic uniform organic organic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
nature “bad” “good” “bad” “good” “good” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
name mun mun dun don dom
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(+) 1/2 1/2 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
sums 1/2 1/2 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(+, +)  3/8  3/8  1/2  1/2  1/2
(−, +) ∗  1/8  1/8  1/2  1/2  1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
sums 1/2 1/2 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(+, +, +)  5/16  5/16  1/3  1/3  1/3
(−, +, +)  1/16  5/48  1/6  1/6  1/6
(+, −, +) ∗  1/16  1/48  1/6  1/6  1/6
(−, −, +)  1/16  1/16  1/3  1/3  1/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
sums 1/2 1/2 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(+, +, +, +)  35/128 35/128  1/4  1/4  1/4
(−, +, +, +)  5/128 35/384  1/12  5/48  1/8
(+, −, +, +)  5/128 7/384  1/12  1/24  0
(−, −, +, +)  3/128 7/128  1/12  5/48  1/8
(+, +, −, +)  5/128 1/128  1/12  5/48  1/8
(−, +, −, +) ∗  3/128 1/384  1/12  1/24  0
(+, −, −, +)  3/128 5/384  1/12  5/48  1/8
(−, −, −, +)  5/128 5/128  1/4  1/4  1/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
sums 1/2 1/2 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(+, +, +, +, +)  63/256 63/256  1/5  1/5  1/5
(−, +, +, +, +)  7/256 21/256  1/20  37/480  1/10
(+, −, +, +, +)  7/256 21/1280  1/20  11/480  0
(−, −, +, +, +)  3/256 63/1280  1/30  1/16  1/10
(+, +, −, +, +)  7/256 9/1280  1/20  11/480  0
(−, +, −, +, +)  3/256 3/1280  1/30  1/120  0
(+, −, −, +, +)  3/256 3/256  1/30  7/240  0
(−, −, −, +, +)  3/256 9/256  1/20  37/480  1/10
(+, +, +, −, +)  7/256 1/256  1/20  37/480  1/10
(−, +, +, −, +)  3/256 1/768  1/30  7/240  0
(+, −, +, −, +) ∗  3/256 1/3840  1/30  1/120  0
(−, −, +, −, +)  3/256 1/1280  1/20  11/480  0
(+, +, −, −, +)  3/256 7/1280  1/30  1/16  1/10
(−, +, −, −, +)  3/256 7/3840  1/20  11/480  0
(+, −, −, −, +)  3/256 7/768  1/20  37/480  1/10
(−, −, −, −, +)  7/256 7/256  1/5  1/5  1/5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
sums 1/2 1/2 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .
name mun mun dun don dom
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .
(+, +, +, +, +, +) 231/1024 231/1024 1/6 1/6 1/6
(−, +, +, +, +, +)  21/1024 77/1024 1/30 71/1152 1/12
(+, −, +, +, +, +) 21/1024 77/5120 1/30 23/1440 0
(−, −, +, +, +, +)  7/1024 231/5120 1/60 29/640 1/12
(+, +, −, +, +, +)  21/1024 33/5120 1/30 11/960  0
(−, +, −, +, +, +)  7/1024 11/5120 1/60 23/5760  0
(+, −, −, +, +, +)  7/1024 11/1024 1/60 47/2880  0
(−, −, −, +, +, +)  5/1024 33/1024 1/60 29/640 1/12
(+, +, +, −, +, +)  21/1024 11/3072 1/30 23/1440  0
(−, +, +, −, +, +)  7/1024 11/9216 1/60 11/1920  0
(+, −, +, −, +, +)  7/1024 11/46080 1/60 1/720  0
(−, −, +, −, +, +)  5/1024 11/15360 1/60 23/5760  0
(+, +, −, −, +, +)  7/1024 77/15360 1/60 47/2880  0
(−, +, −, −, +, +)  5/1024 77/46080 1/60 11/1920  0
(+, −, −, −, +, +)  5/1024 77/9216 1/60 13/576  0
(−, −, −, −, +, +)  7/1024 77/3072 1/30 71/1152 1/12
(+, +, +, +, −, +)  21/1024 7/3072 1/30 71/1152 1/12
(−, +, +, +, −, +)  7/1024 7/9216 1/60 13/576  0
(+, −, +, +, −, +)  7/1024 7/46080 1/60 11/1920  0
(−, −, +, +, −, +)  5/1024 7/15360 1/60 47/2880  0
(+, +, −, +, −, +)  7/1024 1/15360 1/60 23/5760  0
(−, +, −, +, −, +) ∗  5/1024 1/46080 1/60 1/720  0
(+, −, −, +, −, +)  5/1024 1/9216 1/60 11/1920  0
(−, −, −, +, −, +)  7/1024 1/3072 1/30 23/1440  0
(+, +, +, −, −, +)  7/1024 3/1024 1/60 29/640 1/12
(−, +, +, −, −, +)  5/1024 1/1024 1/60 47/2880  0
(+, −, +, −, −, +)  5/1024 1/5120 1/60 23/5760  0
(−, −, +, −, −, +)  7/1024 3/5120 1/30 11/960  0
(+, +, −, −, −, +)  5/1024 21/5120 1/60 29/640 1/12
(−, +, −, −, −, +) 7/1024 7/5120 1/30 23/1440 0
(+, −, −, −, −, +)  7/1024 7/1024 1/30 71/1152 1/12
(−, −, −, −, −, +) 21/1024 21/1024 1/6 1/6 1/6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .
sums 1/2 1/2 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .

8.7 Pinpointing the difference between “good” and “bad”.

We first set some notations :

cnb(r_{1}, r_{2}) := ∑_{k=0}^{k=r_{2}} (−1)^{k} \frac{r_{2}!}{k! (r_{2} − k)!} (r_{2} − k)^{r_{1}}  (232)

In particular :

cnb(r, 1) = 1  (233)
cnb(r, r) = r!  (234)
cnb(r, r_{⋆}) ≥ 2 if 1 < r_{⋆} < r  (235)
cnb(r, r_{⋆}) = 0 if r < r_{⋆}  (236)

Then we set :

f(r_{0}, r_{1}) = ∑_{r_{2}=1}^{r_{2}=r_{1}} f(r_{0} + r_{2}) cnb(r_{1}, r_{2})  (237)

Next, let ω_{r_{0},r_{1}}^{≺ ≺} (resp. ω_{r_{0},r_{1}}^{≻ ≻}) be the arborescent (resp. antiarborescent) sequence obtained by suffixing (resp prefixing) the totally non-ordered sequence (ω′_{1}, …, ω′_{r_{1}}) to the totally ordered sequence (ω_{1}, …, ω_{r_{0}}).

Assume now that F^{•} is some constant-type mould like tu_{a}^{•} (§6.1.4 supra), ie a mould whose values depend solely on the sequence length r, so that F^{ω_{1},...,ω_{r}} ≡ f(r). In view of what precedes, it is clear that after a contracting arborification or antiarborification we get :

F^{ω{≺ ≺ \atop r_{0},r_{1}}} ≡ F^{ω{≻ ≻ \atop r_{0},r_{1}}} ≡ f(r_{0}, r_{1})  with f(r_{0}, r_{1}) as in (284)  (238)

If we take F^{•} := tu_{a}^{•} with a ∈ ℤ, then tu_{a}^{•} is well-behaved and indeed we can see (trivially for a < 0, less so for a > 0) that:

lim sup_{r_{1} → +∞} (\frac{log|f(r_{0}, r_{1})|}{r_{0} + r_{1}}) < +∞  (∀ r_{0} fixed and ≥ 1)  (239)

But if a ∉ ℤ, then tu_{a}^{•} is not well-behaved and we can show that:

lim sup_{r_{1} = +∞} (\frac{log|f(r_{0}, r_{1})|}{r_{0} + r_{1}}) = +∞  (∀ r_{0} fixed and ≥ 1)  (240)

and in fact :

lim sup_{r_{1} → +∞} (\frac{log|f(r_{0}, r_{1})|}{(r_{0} + r_{1}) log(r_{0} + r_{1})}) > 0  (∀ r_{0} fixed and ≥ 1) (241)

9 Proper notion of ramified-exponential growth.

Four cases of increasing difficulty have to be considered, corresponding to four types of resurgence algebras :

(1) the trivial, singularity-free case.

(2) the case with bounded singularities.

(3) the case with ‘manageable’ singularities.

(4) the case with absolutely arbitrary singularities.

In the trivial, singularity-free case, ie for minors that have no singularity other than 0_{•}, the only sensible definition of exponential growth is :

|φ̂(ζ)| ≤ C_{θ , φ} exp (D_{θ , φ} |ζ|) (for ζ large)  (242)

with θ-continuous constants C_{θ , φ} , D_{θ , φ}. The latter continuity condition is essential to ensure uniform exponential growth inside sectors of finite aperture. Clearly, some uniformity condition of this sort will have to be there in the other cases also but, as we saw in §6.5, the simple imposition of bounds, exponential or otherwise, on single paths, won’t do^{90}: some subtle, path-to-path compensation conditions must also come into play. The simplest way to express these constraints is dually to the well-behaved averages: see §9.1.

9.1 Bounded singularities.

This is the case when singularities at the origin are of integrable type and when, close to any singular point ω, the minors remain bounded on any sector of finite aperture and with apex ω.

Let us fix some well-behaved average m – preferably the organic average - – and let us denote m_{θ} its action on the half-axis arg ζ = θ of the Borel plane. If we then define ramified-exponential growth as the existence of finite bounds^{91} (243) (resp (243)+(244)) :

90Indeed, the diemma is this : either we impose uniform exponential bounds (242) on all paths, and then such simple functions as B̂ in §6.1 won’t verify these bounds, or we relax the growth condition on oft-crossing paths in such as way as to accommodate functions like B̂, and in that case it is easy to see that there are going to be functions φ̂ that meet those relaxed conditions and yet cannot be Laplace-summed over ℝ^{+} whichever realness preserving averages m φ̂ we choose to consider. So we are in a fix here, unless we reconcile ourselves to the idea of restricting the growth, not of single determinations φ̂(ζ) , but of suitable averagings of several determinations φ̂(ζ_{i}), for points ζ_{i} with identical projections on ℂ_{•}.

|m_{θ} φ̂(ζ)| ≤ C_{θ , φ} exp (D_{θ , φ} |ζ|)  (243)
|m_{θ} Δ_{ω_{r}} … Δ_{ω_{1}} φ̂(ζ)| ≤ C_{θ , ω, φ} exp (D_{θ , ω, φ} |ζ|)  (244)

then the growth notion so defined meets all four conditions R1, R2, R3, R4.

9.2 Manageable singularities.

A singularity at a point ω ∈ ℂ_{•} of argument arg ω = θ is said to be “manageable” if, by choosing a suitably slow time z_{θ} within the critical time class {z}, the singularity in question can be smoothened (ie rendered bounded, even C^{∞}, or even smoother, but always short of cohesive^{92}) on a whole closed, notched disk centered at ω but whose right and left radii [ω, ω + r e^{iθ}]^{+} and [ω, ω + r e^{iθ}]^{−} have to be regarded as distinct.

Manageable singularities (which can be extremely violent) cover all singularities liable to occur in practice, and in particular all those one encounters in the construction of analysable functions. For resurgent functions with such singularities, ramified-exponential growth is still defined by the existence of bounds (243) or (244), but relative to slow times z_{θ} that may depend on θ and with the usual continuity conditions on θ.

9.3 Arbitrary singularities. An open question.

That leaves only the question of arbitrary, ie possibly unmanageable singularities. Not only do these never occur in natural situations, but it is not at all clear whether they permit a satisfactory, convolution-stable notion of ramified-exponential growth. This, at any rate, would presuppose, among other things, that the implication (248) infra holds for all finite sums (247) obtained by convoluting resurgent functions A^{⋄}_{i} whose minors are regular (say, sectorially bounded at ∞) away from 0_{•} but which may possess horrendous singularities at 0_{•}, and functions B^{⋄}_{i}, whose singular points ω_{i} are ≠ 0_{•} and possibly quite dense, but with singularities there that are fairly tame (say, integrable, or sectorially bounded, or simple poles), and with uniform exponential growth at ∞.

91with constants C_{θ , φ} , D_{θ , φ} or C_{θ , ω , φ} , D_{θ , ω , φ} continuous in θ and ω.

92see [E7] , Lectures 3 and 4.

A^{⋄}_{i} = {Ǎ_{i}, Â_{i}} with Â_{i} regular-bounded away from 0_{•} (245)
B^{⋄}_{i} = {B̌_{i}, B̂_{i}} with B̂_{i} of integrable-exponential type. (246)
C^{⋄} := A^{⋄}_{1} ∗ B^{⋄}_{1} + ⋯ A^{⋄}_{n} ∗ B^{⋄}_{n}  (247)

{C^{⋄}= 0} ⟹^{?} {C = 0}  (248)

That, however, appears to be an open question, even for such quite simple pairs as :

\begin{aligned}Ǎ_{i}(ζ) & := & ∑_{0≤n,m<c.n} a_{i,n,m} ζ^{−n} (log ζ)^{m} \\ B̂_{i}(ζ) & := & ∑_{0≤n} b_{i, ω_{i}} (1 − \frac{ζ}{ω_{i}}) log(1 − \frac{ζ}{ω_{i}}) \end{aligned}

or even :

\begin{aligned}Ǎ_{i}(ζ) & := & ∑_{0≤n} a_{i,n} ζ^{−n} \\ B̂_{i}(ζ) & := & ∑_{0≤n} b_{i, ω_{i}} (ζ − ω_{i})^{−1} \end{aligned}

10 Well-behaved resurgence monomials.

10.1 Reminder about the standard resurgence monomials.

Assume for a start that all ω_{i} are > 0 and consider once again the Δ-friendly hyperlogarithmic monomials ^{±}Ue^{•}(z), but equipped with their exponential factor e^{∥ω∥.z} and expressed in the bases orthogonal to the right/left lateral alien operators Δ Δ_{ω}^{±}. In other words :

Δ Δ_{ω_{0}}^{±} ^{±}Ue^{ω_{1},...,ω_{r}}(z) = ^{±}Ue^{ω_{2},...,ω_{r}}(z) if ω_{0} = ω_{1}
= 0  if ω_{0} ≠ ω_{1}

Let us re-write the integral (106) for these monomials, but in a suggestive, easily generalisable form. Setting go_{ω}(y) := e^{−ω y} we get :

+Ue^{ω_{1},..., ω_{r}}(z) := \frac{1}{(2πi)^{r}} ^{+} ∫_{0}^{∞} \frac{go_{ω_{1}}(y_{1}) … go_{ω_{r}}(y_{r})}{go_{ω_{1}}(z) … go_{ω_{r}}(z)} \frac{dy_{1} … dy_{r}}{(y_{r} − y_{r−1}) … (y_{1} − z)}

−Ue^{ω_{1},..., ω_{r}}(z) := \frac{1}{(2πi)^{r}} ^{−} ∫_{0}^{∞} \frac{go_{ω_{1}}(y_{1}) … go_{ω_{r}}(y_{r})}{go_{ω_{1}}(z) … go_{ω_{r}}(z)} \frac{dy_{1} … dy_{r}}{(y_{r} − y_{r−1}) … (y_{1} − z)}

with lateral integration right (resp left ) of arg ζ = 0 in the first (resp second) integral^{93}.

10.2 Well-behaved resurgence monomials induced by prodiffusions.

Instead of pairs {f_{ω}, g_{ω}} connected by the Fourier transform, we are now dealing we pairs {fo_{ω}, go_{ω}} connected by the Borel-Laplace transform :

fo_{ω}(x) = ∫_{0}^{∞} e^{−x y} go_{ω}(y) dy  (249)
go_{ω}(x) = \frac{1}{2πi} ∫_{−i∞}^{+i∞} e^{+x y} fo_{ω}(x) dx  (250)

We still have to consider multiplicative semigroups {go_{ω}(•) , ω > 0} and convolution semigroups {fo_{ω}(•) , ω > 0} but the latter are now relative to the vertical, open-ended convolution ◁▷ :

go_{ω}(y) ≡ (go_{1}(y))^{ω}  ∀ω  (go_{1}(y) > 0)  (251)
fo_{ω_{1}}◁▷fo_{ω_{2}} ≡ fo_{ω_{1}+ω_{2}}  ∀ω_{i}  with  (252)
(fo_{ω_{1}}◁▷fo_{ω_{2}})(x) := \frac{1}{2πi} ∫_{−i∞}^{+i∞} fo_{ω_{1}}(x_{1}) fo_{ω_{2}}(x − x_{1}) dx_{1}  (253)

The function go_{1}(y) should be analytic on ℂ_{•} or, better still, uniformanalytic on ℂ^{∗}, and it should fulfill too additional conditions at least :

(i) log(go_{1}(y)) ∼ −y as y → +∞ (ii) go_{1}(y)) should be real for arg y = 0 (iii) go_{1}(y) should decrease fast enough as y ↓ +0 for ∫_{−i∞}^{+i∞} |fo_{1}(x)||dx| < ∞ (iv) if go_{ω}(y) = go_{ω,c}(y) has to depend on a parameters c, then there should be invariance under some simple change (y, c, ω) ↦ (l y, l^{−1} ω, l^{k} c) ^{94}.

Condition (i) is there to ensure that the integrals of §10.1 retain the right asymptotic behaviour at ∞ in the geometric model and sum up to proper resurgence monomials orthogonal to the Δ Δ_{ω}^{±} Condition (ii) is there to ensure the ‘realness’ property M2 . Condition (iii), which of course wasn’t verified for the earlier choice of go_{1}(y) := e^{−y} in §10.1, is there to ensure the crucial property M3 of ‘proper growth’ in the ω variables. Condition (iv) is there to ensure the homogeneity property M4 .

93it is the change from Δ Δ-orthogonality to Δ Δ^{±}-orthogonality that is responsible for the appearance of the factor (2πi)^{−r} in front of the integrals.

94∀l > 0.

10.3 Canonical well-behaved resurgence monomials.

The four conditions (i)-(iv) leave a huge freedom of choice, but let us go straight for the simplest and most economical choice, which corresponds to a kernel function :

g_{ω_{0}}(y) := exp(−ωy − c^{2}ω̄y^{−1}) (ω ∈ ℂ^{⋆}; c ≥ 0)  (254)

that involves only one free parameter c > 0 and meets all four conditions above, including (iv) with k = 1.

This leads to the formulas :

Ua_{c}^{ω_{1},...,ω_{r}}(z) := SPA ∫_{0}^{∞} \frac{e^{− ∑_{1}^{r}(ω_{i} y_{i}+c^{2}ω̄_{i} y_{i}^{−1})} dy_{1} … dy_{r}}{(y_{r} − y_{r−1}) … (y_{2} − y_{1})(y_{1} − z)} (255)
Ue_{c}^{ω_{1},...,ω_{r}}(z) := Ua_{c}^{ω_{1},...,ω_{r}}(z) e^{(∑_{1}^{r} ω_{i}) z+(∑_{1}^{r} ω̄_{i}) c^{2} z^{−1}}  (256)
U_{c}^{ω_{1},...,ω_{r}}(z) := Ua_{c}^{ω_{1},...,ω_{r}}(z) e^{(∑_{1}^{r} ω̄_{i}) c^{2} z^{−1}}  (257)

with integration along the rays arg(ω_{i} y_{i}) = arg(ω̄_{i}/y_{i}) = 0.

The Ua_{c}^{ω} are auxiliary expressions. The resurgence monomials proper are the U_{c}^{ω} (orthogonal to the ordinary alien derivations Δ_{ω}) and the exponentialcarrying Ue_{c}^{ω} (orthogonal to the exponential-carrying alien derivations Δ Δ_{ω}).

Interpretation of “SPA ”. SPA in front of the integral means suitable path average. If we integrate first in y_{1}, then y_{2}, etc, the question arises^{95} as to how (ie on which side) y_{i} should bypass the next (yet unused) variable y_{i+1}. If to the right, we set ϵ_{i} := +. If to the left, we set ϵ_{i} := −. To each choice {ϵ_{1}, ϵ_{2}, … } there corresponds a different integration path, and SPA means that one should take a precise average of all such paths, depending on which system Δ_{ω} of alien derivations one wishes the U^{ω} to be orthogonal to. But for the right- or left-lateral moulds (characterised by orthogonality to Δ_{ω}^{±}) the SPA average reduces to one single path, with all ϵ_{i} identical (either + or −). In that case, however, one should add the factor (2πi)^{−r} in front of the integrals, as in §10.1.

Interpretation of 1/(y_{1} − z) The integral (134) defines U_{c}^{ω} in all three models (formal, geometric, convolutive) at one stroke, depending on how we construe 1/(y_{1} − z) :

– as a power series in z^{−1}, – or as a function germ at ∞,

95at least when two consecutive integration axes coincide, ie when arg ω_{i} = arg ω_{i+1}.

– or again as its own Borel tranform.

\begin{matrix}F ormal model : & \frac{1}{y_{1}−z} → & − ∑_{0}^{∞} z^{−n−1} y_{1}^{n} \\ ⟹ & Ũ_{c}^{ω}(z) & as a formal power series \\ Geometric model : & \frac{1}{y_{1}−z} → & z-germ at ∞ \\ ⟹ & U_{c}^{ω}(z) & as a sectorial z-germ at ∞ \\ Convolutive model : & \frac{1}{y_{1}−z} → & − exp(y_{1} ζ) \\ ⟹ & Û_{c}^{ω}(ζ) & as a full ζ-germ at 0 \end{matrix}

Main result: For positive values c > 0 of the twist, U_{c}^{•} constitutes a well-behaved, multiplicative system of resurgence monomials.

The limit-case c = 0. In the limit-case c = 0 the integrals (134) remain well-defined and we still have a multiplicative system U_{0}^{•} of resurgence monomials, but it is no longer well-behaved. In fact, it coincides with the much more ancient system U^{•} of hyperlogarithmic monomials, so-called because their dependence on the ω_{i}’s is indeed of hyperlogarithmic type. In contradistinction, the U_{c}^{•} and the host of special functions attached to them (see §6.7) are called paralogarithmic.

Why “twisted”, why “spherical”, why “canonical” ? The presence of a free parameter c slightly detracts from the “canonicity” of our system, but this cannot be helped : no system of well-behaved resurgence monomials can suffice for all problems unless there is at least one free parameter that can be adjusted from case to case. The miracle is rather that one parameter should be enough ! So much for the twist. As for “spherical”, it refers to the striking symmetry of behaviour which our monomials U_{c}^{•} exhibit at the antipodes 0 and ∞ of the Riemann sphere when c > 0, and which, remarkably enough, disappears when we “untwist” them, ie when c = 0.

10.4 Proofs and comments.

Let^{±}Ua_{c}^{ω_{1},...,ω_{r}}(z, η) denote the Δ Δ^{±} orthogonal variant of our canonical monomials, in the geometric-sectorial model that corresponds to Borel-Laplace integration right (resp left) of the axis arg ζ = 0 if η = + (resp -). The orthogonality relations to ALIEN can easily be checked on the y-integral representations

ϵUe_{c}^{ω_{1},..., ω_{r}}(z, η) := \frac{(ϵ1)^{r}}{(2πi)^{r}} ^{ϵ,η} ∫_{0}^{∞} \frac{go_{ω_{1}}(y_{1}) … go_{ω_{r}}(y_{r})}{go_{ω_{1}}(z) … go_{ω_{r}}(z)} \frac{dy_{1} … dy_{r}}{(y_{r} − y_{r−1}) … (y_{1} − z)}

with ‘lateral circumvention’ or, if you prefer, with integration on close but distinct axes θ_{i} ∼ 0 :

ϵ (θ − θ_{1}) > 0 ; η (θ_{i} − θ_{i+1}) > 0 (arg z = θ ; arg y_{i} = θ_{i} ; ϵ, η ∈ {+, −})

The finite’ conditions M1, M2, M4 offer no difficulties. M1 is easily checked on the y-integrals. M2 is immediate. So is M4 : its expression in the multiplicative models (geometric or formal) is simply the invariance of the monomials under dilatations (y, c, ω_{i}) ↦ (l y, l c, l^{−1} ω_{i}) , ∀l > 0. In a sense, this is less than the “exact” homogeneousness (namely : invariance under the changes (y, ω_{i}) ↦ (l y, l^{−1} ω_{i}) as with the standard or hyperlogarithmic monomials of §2.4) that we might wish for, but this cannot be helped as soon as a parameter c comes into play. Moreover, this slight ‘defect’ is more than offset by the emergence of a new symmetry – under the antipodal involution , see §11.4, §12.4 – which has no ‘standard’ equivalent.

But the difficult bit is of course the growth condition M3. Let us first use the elementary moulds of §2.1 to rewrite the y-integral and its companion x-integral in a form reminiscent^{96} of the formulas for the lateral moulds of the uniformising averages :

±Ua_{c}^{ω_{1},...,ω_{r}}(z, η) = \frac{(±1)^{r}}{(2πi)^{r}} ∫_{0}^{∞} go_{ω_{1}}(y_{1}) … go_{ω_{1}}(y_{1}) tas_{z,∞}^{y_{1},...,y_{r}} dy_{1} … dy_{r}

= ∓ η \frac{(−1)^{r}}{(2πi)^{r}} ∫_{−i∞}^{+∞} σ_{η}(x̂_{1}) σ_{∓}(x̂_{2}) … σ_{∓}(x̂_{2})
× fo_{ω_{1}}(x_{1}) … fo_{ω_{r}}(x_{r}) e^{∥x∥ z}dx_{1} … dx_{r}

Here, it is more convenient to use the criteria M3.1 or M3.3, which involve contracting arborification. As with the uniformising averages, we have the choice between two strategies.

Either we choose to work with the x-integrals. This means subjecting the moulds antisofo^{•} or antisefo^{•} (depending on η) to contracting arborification and observing that this operation leaves their outward expression unchanged : they are still given by the formulas of §2.1.9, but with sums x̂_{i} := ∑_{i≤j} x_{j} now relative to the arborescent order.

96except that the y-sequence must now be inversed, and the x̌_{i}-sums (22) replaced by x̂_{i}-sums (23).

Or we can work instead with the y-integrals. This means subjecting the mould tas_{z,∞}^{•}:

tas_{z,∞}^{y_{1},...,y_{r}} := ∏_{1≤i≤r} \frac{1}{y_{i_{−}} − y_{i}}  with  (258)
y_{i_{−}} := y_{i−1} if i > 1 , y_{i_{−}} := z if i = 1  (259)

to ordinary arborification (which, due to the circumvention rules, translates into contracting arborification for the integrals) and observe that this operation doesn’t alter the factorisation of §2.1.8, but merely changes the interpretation of y_{i_{−}}: if y_{i} is not a root (minimal element), then y_{i_{−}} must be interpreted as the (unique) antecedent of y_{i} within the arborescent sequence y^{≺}, and if y_{i} is a root , y_{i_{−}} should be set equal to z.

Both methods not only yield the required bounds M3_{1},M3_{3}, but there even appears an important factor e^{−C_{∗} c} , C_{∗} > 0 in front of the constants. Checking the fact of “form-preserving arborification” lemma (see §2.1.11) is easier with the polar mould tas_{z,∞}^{•} than with the flat moulds antisofo^{•} or antisefo^{•}, but with the first method (x-integration) the bounds M3_{1},M3_{3} are immediate to establish, since the ‘flat’ part of the integrand is either 0 or ±1 and each fo_{ω_{i}}(x_{i}) is bounded on i ℝ, whereas with the second method (yintegration) one should carefully select (near 0) the multipath of integration in order to control the smallness of the denominators y_{i_{−}} − y_{i}.

10.5 Extension to the multicritical case.

General monocritical paralogaritms.

In the 6 examples discussed in §3, we have on purpose chosen the simplest normal forms Ob^{nor}, thereby ensuring that all objects Ob ∼ Ob^{nor} could be represented by entire power series (and exponentials). But this doesn’t hold for all conjugacy classes. Take for instance Example 2 in §3.2 and replace the normal form (75) by d_{x} y^{nor} = y^{nor} + σ z^{−1} y^{nor} with σ ∉ ℤ. The formal integral (77) is no longer in ℂ[[z^{−1} , u e^{z}]] but in ℂ[[z^{−1} , u e^{z} z^{σ}]], which means that we are stuck with non-entire powers of z. This has three main consequences :

(i) as far as alien calculus is concerned, the indices ω of our alien derivations must now be regarded as elements of ℂ_{•} rather than ℂ^{∗}.

(ii) as far as object analysis is concerned, the total “quantity” of (independent) analytic invariants doesn’t increase, because for each pair ω, ω′ ∈ ℂ_{•} with the same projections ω̇, ω̇′ ∈ ℂ^{∗}, the corresponding invariants 𝔸_{ω} and 𝔸_{ω′} are simply related (one is deducible from the other).

(iii) as far as object synthesis is concerned, we must now work with the ramified variants of our Δ-friendly twisted monomials, obtained by systematically replacing in all three integrals (255),(256),(257) the elements ϖ(y) := ω y + c^{2} ω̄ y^{−1} by their ramified counterparts ϖ(y) := ω y + c^{2} ω̄ y^{−1} + σ log y. That apart, nothing much changes, the new monomials being every bit as well-behaved as the unramified prototypes.

General polycritical paralogaritms.

In polycritical situations (exemplified by Ex 4 and Ex 6 in §3) object synthesis calls for polycritical twisted Δ-friendly monomials. We construct them by replacing the binomial elements ϖ(y) := ω y + c^{2} ω̄ y^{−1} by the bipolynomial elements :

ϖ(y) := ∑_{p}(ω_{p} y^{p} + c_{p}^{2} ω̄_{p} y^{−p})

or, in ramified situations, by :

ϖ(y) := ∑_{p}(ω_{p} y^{p} + c_{p}^{2} ω̄_{p} y^{−p}) + σ log y

Of course, for each higher critical time class z_{p} := {z^{p}}, the monomials’ expression in the corresponding geometric model (resp Borel plane) involves a polarisation θ_{q} for each lower (resp strictly lower) critical time classe z_{q} := {z^{q}} with q ≤ p (resp q < p).

As for the several twist parameters c_{p} attached to the various classes z_{p}, we may of course collapse them into one by setting c_{p} := c or c_{p} := p c or c_{p} := p^{−1} c etc, but there seems to be no compelling reason for preferring one choice to the other.

∂-friendly paralogarithms.

The thorough symmetry which obtains, in the standard, twistless context, between the Δ-friendly and ∂-friendly monomials U^{•} and V^{•} and their monics U^{•} and V^{•}, and which comes through graphically in the diagram of §4.1 and the formulas of §12.2.2 – that symmetry does survive for the twisted monomials, but in less perfect a manner. Some of the relevant formulas are mentioned in §12.3.6. It should be noted, however, that the ∂-friendly monomials serve no purpose in object analysis (the standard, twistless monomials fully suffice there). Their raison d’ être is rather :

(i) to lead to a better understanding of the twisted Δ-friendly monomials (ii) to lead to simple formulas for the corresponding monics U^{•} and V^{•}.

The corresponding developments would fill too much space here, but one curious feature perhaps deserves mention : whereas the classical gamma function Γ(σ) := ∫_{0}^{∞} e^{−t} t^{σ−1} dt totally dominates the subject of ramified hyperlogarithms , for the ramified paralogarithms it should be replaced by a twisted look-alike Γ_{c}(σ) := ∫_{0}^{∞} e^{−t−c^{2} t^{−1}} t^{σ−1} dt whose properties, however (entireness, asymptotics etc) couldn’t be more different as soon as c > 0.

11 Applications to canonical Object Synthesis.

11.1 Outline of the construction.

Basically, with the twisted monomials at our disposal, Object Synthesis becomes a purely mechanical affair. This is precisely what we had set out to achieve : to reduce the whole process to a succession of formal manipulations. We saw in §4 why the early attempts, based on the standard monomials U^{•}(z), couldn’t fully succeed. But with their twisted counterparts U_{c}^{•}(z) , everything works fine.

Let us outline the six main steps :

Step 1: select a formal class of local analytic objects, characterised by a formal normal form Ob^{nor}, and start from any given admissible system of analytic invariants 𝔸 = {𝔸_{ω}; ω ∈ Ω}

Step 2: choose a well-behaved system of alien derivations, preferably the organic system Δ^{org} = {Δ_{ω}^{org} ; ω ∈ ℂ^{⋆}} , and express the analytic invariants in the corresponding basis 𝔸^{org} = {𝔸_{ω}^{org} ; ω ∈ Ω}

Step 3: solve “mechanically”^{97} the system of resurgence equations that characterise the direct or inverse normaliser Θ^{±1}. For instance, in the case of simply resonant fields we find these expansions^{98} :

97ie without worrying about convergence. Mark the choice of words: mechanically, ie by means of expansions into series (127) of abstract resurgence monomials, rather than formally, which would suggest solving the problem in the ring of formal power series. To bring out the difference, we might also, as in §4, speak of semi-formal expansions.

Θ  :=^{always} 1 + ∑_{1≤r} ∑_{ω_{i}∈Ω}(−1)^{r} Ue_{org}^{ω_{1},...,ω_{r}}(z) 𝔸_{ω_{r}}^{org} … 𝔸_{ω_{1}}^{org} (260)
Θ^{−1} :=^{conditionally} 1 + ∑_{1≤r} ∑_{ω_{i}∈Ω}  Ue_{org}^{ω_{r},...,ω_{1}}(z) 𝔸_{ω_{r}}^{org} … 𝔸_{ω_{1}}^{org} (261)

Step 4: replace in that “mechanical” solution the abstract monomials U_{org}^{ω}(z) by the twisted or spherical monomials U_{c , org}^{ω}(z) for a large enough twist c.

Step 5: arborify the expansions according to the rules given in §6.1 so as to render the previously obtained expansions for Θ^{±1} convergent in the space of resurgent functions.

Step 6: Construct the sought-after analytic object Ob from its normaliser by using Ob = Θ Ob^{nor} Θ^{−1}

The reader may easily work this out in the case of our four monocritical examples of §3 and §4 (Ex 1 through 4). The polycritical examples (Ex 5 and 6) also respond to the same treatment, except that acceleration theory^{99} is needed. Here, to avoid drowning in secondary details, we shall focus only on the central point, namely the convergence, after arborification, of the mechanical expansions into series of twisted monomials.

11.2 Proof of the convergence for a positive or large enough twist c.

Generic divergence prior to arborification.

Let us work with the direct normalisers. Prior to arborification, their expansions, whether we write them in a well-behaved basis, say the organic basis^{100} :

Θ = ∑_{ω}Ue_{c}^{ω}(z) 𝔸_{ω} = ∑_{r} ∑_{ω_{i}} Ue_{c}^{ω_{1},...,ω_{r}}(z) 𝔸_{ω_{r}} … 𝔸_{ω_{1}}  (262)

98The second expansion, for the reverse normaliser, is valid only if all the invariants 𝔸_{ω} have no ∂_{z}-component and so commute with the resurgence monomials Ue^{ω}(z). When this is not the case, the expansion (261) should be slightly modified, but one can also be content to work with (260), which is always valid, and then derive Θ^{−1} by straightforward inversion of Θ.

99see for instance [E4][E6][E7].

100for brevity we drop the indes “org” in the sequel.

or in one of the (right/left) lateral bases :

Θ = ∑_{ω}^{±}Ue_{c}^{ω}(z) 𝔸_{ω}^{±} = ∑_{r} ∑_{ω_{i}} ^{±}Ue_{c}^{ω_{1},...,ω_{r}}(z) 𝔸_{ω_{r}}^{±} … 𝔸_{ω_{1}}^{±}  (263)

are hopelessly divergent^{101} since we cannot expect better bounds than :

∥Ue_{c}^{ω}∥ ≤ c_{1}^{r} e^{−C_{0} c r}  ;  ∥𝔸_{ω}∥ ≤ c_{2}^{r} r!  (0 < c_{i} < ∞)(264)
∥^{±}Ue_{c}^{ω}∥ ≤ c_{3}^{r} e^{−C_{0} c r}  ;  ∥𝔸_{ω}^{±}∥ ≤ c_{4}^{r} r!  (0 < c_{i} < ∞)(265)

r being the length of the totally ordered sequence ω. So we have to arborify these expansions:

∑_{ω}Ue_{c}^{ω}(z) 𝔸_{ω} → ∑_{ω^{≺}} Ue_{c}^{ω^{≺}}(z) 𝔸_{ω}^{≺}  (ordinary arborification)
∑_{ω}^{±}Ue_{c}^{ω}(z) 𝔸_{ω}^{±} → ∑_{ω≺ ≺}^{±}Ue_{c}^{ω^{≺ ≺}}(z) 𝔸_{ω^{≺ ≺}}^{±} (contracting arborification)

Under the dual definitions for arborification and coarborification (see §2.1.4), the operation leaves our expansions for Θ formally unchanged : it merely rearranges their terms. But instead of the unsatisfactory bounds (264), (265) we shall now get new ones :

∥Ue_{c}^{ω^{≺}}∥ ≤ C_{1}^{r} e^{−C_{0} c r}  ;  ∥𝔸_{ω}^{≺}∥ ≤ C_{2}^{r}  (0 < C_{i} < ∞)(266)
∥^{±}Ue_{c}^{ω^{≺ ≺}} ∥ ≤ C_{3}^{r} e^{−C_{0} c r}  ;  ∥𝔸_{ω^{≺ ≺}}^{±} ∥ ≤ C_{4}^{r}  (0 < C_{i} < ∞)(267)

that shall ensure absolute convergence.

The removability of the factor r! in the comould part doesn’t come as a great surprise, since the coarborification constraints^{102} split each 𝔸_{ω} into a sum of roughly r! terms. But we might fear, dually, the appearance of a factor r! in the mould part, since the arborification rules^{103} regroup roughly r! terms on the right-hand side. This is indeed what arborification does to a ‘random’ mould : it creates a factor of magnitude r!. However, most applications involve moulds which are far from ‘random’ : not only do they fall into one of the four basic symmetry types, but they usually verify additional identities, which ensure that they retain their ‘size’ (and sometimes even their ‘form’) under arborification.

Arborification here doesn’t increase mould norms. This applies to most of the moulds catalogued in §2.1, with only a handful of exceptions that are mentioned there^{104}. This applies in particular, as we showed in §10.4, to the twisted resurgence monomials, as soon as their twist c is > 0.

101generically – except in linear problems like Example 3.

102see §2.1.4.

103see §2.1.4.

104see also §8.7.

Homogeneous coarborification.

Let {𝔹_{ω}, ω ∈ Ω} be any system of ordinary differential operators in the variables x_{1}, …, x_{ν} and define the comould 𝔹_{•} as usual by setting :

𝔹_{ω_{1},..., ω_{r}} := 𝔹_{ω_{r}} … 𝔹_{ω_{1}}  (268)

Then there exists a privileged arborescent comould 𝔹_{•^{≺}} , the so-called homogeneous co-arborification of 𝔹_{•} which is entirely characterised by the following three properties : P1 𝔹_{•^{≺}} is co-separative^{105}, ie :

𝔹_{ω}^{≺} (φ_{1} φ_{2}) ≡ ∑_{ω^{1}^{≺}⊕ω^{2}^{≺}=ω^{≺}} 𝔹_{ω^{1}}_{≺} (φ_{1}) 𝔹_{ω^{2}}_{≺} (φ_{2})  (269)

P2 If deg(ω^{≺}) = d ie if the tree ω^{≺} has exactly d roots, then the operator is homogeneous in the ∂_{i} := ∂_{x_{i}} with total degree d P3 If ω = ω_{1} ω^{∗} (in other words, if ω is of degree one, with a root element ω_{1} followed by some arborescent sequence ω^{∗}^{≺}) the corresponding operator factors as :

𝔹_{ω}^{≺} x_{j} ≡ 𝔹_{ω^{∗}}^{≺} 𝔹_{ω_{1}} log x_{j}  (j = 1, 2, …, ν)  (270)

Moreover, if 𝔹_{•} is co-symmetral ^{106} (resp co-symmetrel ^{107}), then 𝔹_{•} and 𝔹_{•}^{≺} are indeed correlated according to B_{ω} := ∑_{ω^{≺}<ω} B_{ω}^{≺} (resp B_{ω} := ∑_{ω^{≺ ≺} <<ω} B_{ω}^{≺ ≺} ). In other wordss, whereas symmetral and symmetrel moulds obey different arborification rules (simple versus contracting), the homogeneous co-arborification rules are exactly the same for a co-symmetral comould (like 𝔸_{•} above) and a co-symmetral comould (like 𝔸±_{•} above).

Let us check, by induction on the length r of ω^{≺}, the fact that P1, P2, P3 together do determine 𝔹_{ω}^{≺} .

105 ω^{1}^{≺} ⊕ ω^{2}^{≺} denotes the tree obtained by juxtaposition of ω^{1}^{≺} and ω^{2}^{≺}, with no other order relations than those inherited from the sub-trees ω^{i}. The sum (279) extends also to the trivial juxtapositions, with one summand ω^{i}^{≺} equal to ω^{≺} and the other one empty.

106see (17).

107see(17).

Either d(ω^{≺}) = 1, which means that ω^{≺} is of the form (271), in which case 𝔹_{ω}^{≺} is as below :

ω^{≺} = (ω_{1}, ω^{∗}^{≺})  (271)
𝔹_{ω}^{≺} = ∑_{1≤i≤ν}(𝔹_{ω^{∗}}^{≺}.𝔹_{ω_{1}}^{≺}. log x_{j})(x_{j} ∂_{j})

Or deg(ω^{≺}) = d ≥ 2, which means that ω^{≺} is of the form (272), with s clusters of d_{1}, …, d_{s} identical, irreducible summands ω^{i_{1}}^{≺}, …, ω^{i_{s}}^{≺}, in which case 𝔹_{ω}^{≺} is as below :

ω^{≺} = ω^{1}^{≺} ⊕ ⋯ ⊕ ω^{d}^{≺}  (ω^{i}^{≺} ≠ ∅ , deg(ω^{i}^{≺}) = 1)
= (ω^{i_{1}}^{≺})^{⊕d_{1}} ⊕ ⋯ ⊕ (ω^{i_{s}}^{≺})^{⊕d_{s}}  (d_{1} + ⋯ + d_{s} = d)  (272)
𝔹_{ω}^{≺} = \frac{1}{d_{1}! … d_{s}!} ∑_{{1≤s≤d \atop 1≤js≤ν}} (𝔹_{ω_{1}}^{≺}. log x_{j_{1}}) … (𝔹_{ω_{d}}^{≺}. log x_{j_{d}}) (x_{j_{1}} ∂_{j_{1}}) … (x_{j_{d}} ∂_{j_{d}})

Coarborification diminishes comould norms.

The phenomenon takes place for any reasonable norm on local differential operators, for instance :

∥𝔹∥ = ∥𝔹∥_{D_{1},D_{2}} := sup_{φ≠0} \frac{∥𝔹 φ∥_{D_{1}}}{∥φ∥_{D_{2}}}  with 0 ∈ D_{1} , D̄_{1} ⊂ D_{2} ⊂ ℂ^{ν} (273)

with D_{1}, D_{2} two small open neighbourhoods of 0 and ∥φ∥_{D_{i}} the uniform norm on D_{i}. To illustrate norm reduction, ie the improvement from (274) to (275) :

∥𝔹_{ω}∥ ≤ r(ω^{≺})! C^{N(ω^{≺})} ∥𝔹_{ω_{1}}∥ … ∥𝔹_{ω_{r}}∥  (274)
∥𝔹_{ω}^{≺}∥ ≤  C^{N(ω^{≺})} ∥𝔹_{ω_{1}}∥ … ∥𝔹_{ω_{r}}∥  (275)

let us fix a non-resonant spectrum λ ∈ ℂ^{ν} and consider first-order differential operators of the form :

𝔹_{ω_{i}} := x^{n_{i}} B_{ω_{i}}  with B_{ω_{i}} := ∑_{1≤j≤ν} B_{ω_{i}}^{j} x_{j} ∂_{x_{j}} , ω_{i} :=< n_{i}, λ > (276)

Next, let us carry out homogeneous coarborification for three extreme types of arborescent sequences :

ω  := (ω_{1}, …, ω_{r})  ; total order ; all ω_{i} distinct
ω′^{≺} := (ω_{1}, …, ω_{r})  ; total order ; all ω_{i} distinct
ω′′^{≺} := (ω_{r} ⊕ ⋯ ⊕ ω_{1}) ; no order ; all ω_{i} distinct
ω′′′^{≺} := (ω_{r} ⊕ ⋯ ⊕ ω_{1}) ; no order ; all ω_{i} identical

We find :

𝔹_{ω}  := 𝔹_{ω_{r}} … 𝔹_{ω_{1}}  (277)
𝔹_{ω′}^{≺} := x^{n_{r}}(B_{ω_{r}} x^{n_{r−1}}) (B_{ω_{r−1}} x^{n_{r−2}}) … (B_{ω_{3}} x^{n_{2}}) (B_{ω_{2}} x^{n_{1}})B_{ω_{1}} (278)
𝔹_{ω′′}^{≺} := x^{n_{1}+···+n_{r}} B_{ω_{1}} … B_{ω_{r}}  (279)
𝔹_{ω′′′}^{≺} := \frac{1}{r!} x^{n_{1}+···+n_{r}} B_{ω_{1}} … B_{ω_{r}}  (280)

and in all three cases we observe the disappearance of the factor r! , though for rather distinct reasons :

– in (278) we have a first-order differential operator B_{ω_{1}} preceded by innocuous scalar factors B_{ω_{i}} x^{n_{i−1}}

– in (279) we a differential operator B_{ω_{1}} … B_{ω_{r}} (all terms commute) of order r and of factorially large norm, but with a more than factorially small front factor x^{∥n∥} since ∥n∥ ≥ const . r^{1+ \frac{1}{ν}}

– in (280) we have again a differential operator B_{ω_{1}} … B_{ω_{r}} (all terms are equal) of order r and of factorially large norm, but with a multiplicity factor \frac{1}{r!} in front.

In fact, norm reduction holds for all arborescent sequences : see for instance [E5],§4.

Convergence after twisting and arborification.

At this stage, to establish the normal convergence of the arborified expansions for the direct normalisers Θ for a large enough value of the twist c, it is enough :

– to pair the mould-comould estimates in (266) or (267)

– to establish the existence of bounds q(N) ≤ Q_{0} Q_{1}^{N} where q(N) denotes the total number of arborescent sequences ω^{≺} = (ω_{i})^{≺} corresponding to multiinteger sequences n^{≺} = (n_{i})^{≺} with n_{i} ∈ ℕ^{ν} of sum ∥n^{≺}∥ = ∑ |n_{i}| = N ^{108}.

The one-sided/two-sided dichotomy.

Actually, the choice of a large enough twist c is necessary only in what we might call problems of two-sided resurgence (which, in the context of our 6 examples, would cover the binary, sesquilateral, and bilateral cases) ie in all problems where the index set Ω is such that in the expansions (262) infinitely many ω = (ω_{i}) may contribute to the same total frequency ω_{0} = ∥ω∥ = ∑ ω_{i}.

108one can take Q_{1} := 4.2^{ν} , see [E5],pp94-95.

But in problems of one-sided resurgence (in the context of our 6 examples, this would cover the unary and unilateral cases) we have convergence for any c > 0.^{109}

The linear/non-linear dichotomy.

On the other hand, the recourse to arborification is indispensible only for non-linear problems. For linear or affine problems (such as Example 3 and 5), the unarborified expansions are already (normally) convergent. Lastly, in the very exceptional – and elementary – instance of problems that are both linear/affine and one-sided (this would correspond to the unary subcase in Ex 1 and Ex 2) there is need for neither twist nor arborification.

The total picture can be neatly summed up in the following table :

linear problems  non-linear problem
one-sided resurgence  c ≥ 0 ; without arbor. c > 0 ; with arbor.
two-sided resurgence  c ≫ 1 ; without arbor. c ≫ 1 ; with arbor.

11.3 What is so special about the twistless case ?

Basically, we already know the answer : only for c > 0 are the “prodistribution” functions fo_{ω}(x) absolutely integrable at x = ±i∞, and this fact in turn is responsible for property M3. But we must now look concretely at the difference which a positive c makes to object synthesis. Let us reason on Example 2 (§3.2) and focus on the unilateral case (see §4.5.3,§4.6) because, unlike the bilateral case , it allows us to take c as small as we wish.

Starting from some admissible unilateral system {𝔸_{n} := u^{n+1}∂_{u} , n ∈ ℕ^{+}} of analytic invariants, using the mould tas_{a,b}^{•} of §2.1.8, and taking recourse to the usual SPA trick (special path averaging), let us form the general substitution operators :

H_{b,a} := (2πi)^{−r} SPA∫_{0}^{∞} tas_{a,b}^{y_{1},...,y_{r}} e^{− ∑_{1≤i≤r}(n_{i}y_{i}+c^{2}n_{i} y_{i}^{−1})} ×
dy_{1} … dy_{r} v^{n_{1}+···+n_{r}} 𝔸_{n_{r}} … 𝔸_{n_{r}} (281)

which verify^{110} :

H_{b,a} φ = φ ∘ h_{a,b}  with  h_{a,b} := H_{b,a}.u  (282)
H_{a_{3},a_{2}} . H_{a_{2},a_{1}} = H_{a_{3},a_{1}} but h_{a_{1},a_{2}} ∘ h_{a_{2},a_{3}} = h_{a_{1},a_{3}}  (283)

109but usually not for c = 0. See §4.6.

110note the index exchange H_{b,a} → h_{a,b} which reflects the composition rules

and which, for suitable specialisations of the parameters a, b, v, yield the direct and inverse normalisers :

\begin{aligned}Θ & ≡ & H_{∞,z} ,  θ & ≡ & h_{z,∞}  for & v := e^{z+c^{2} z^{−1}} \\ Θ^{−1} & ≡ & H_{z,∞} ,  θ^{−1} & ≡ & h_{∞,z}  for & v := e^{z+c^{2} z^{−1}} \end{aligned}

For instance, if we set all invariants equal to 0 and retain only the first one, ie 𝔸_{1} = A_{1} u^{2} ∂_{u} ≠ 0, we get:

h_{a,b}  = \frac{u}{1+A_{1} ( Ua_{c}^{1}(a)−Ua_{c}^{1}(b)) v u}  ( for 𝔸_{1} = A_{1} u^{n+1} ∂_{u} )
θ(u) = \frac{u}{1+A_{1} Ua_{c}^{1}(z) e^{z+c^{2} z^{−1}} u} = \frac{u}{1+A_{1} Ue_{c}^{1}(z) u}
θ^{−1}(u) = \frac{u}{1−A_{1} Ua_{c}^{1}(z) e^{z+c^{2} z^{−1}} u} = \frac{u}{1−A_{1} Ue_{c}^{1}(z) u}

Let us now add infinitesimal, admissible perturbations δ𝔸_{n} (n ≥ 2), thus changing our invariant system {𝔸_{1}, 0, 0, … } to {𝔸_{1}, δ𝔸_{2}, δ𝔸_{3}, … }. Since :

tas_{a,b}^{y_{1},...,y_{r}} ≡ tas_{a,y_{i}}^{y_{1},...,y_{i−1}} \frac{a − b}{(a − y_{i}) (y_{i} − b)} tas_{y,b}^{y_{i+1},...,y_{r}} (∀i, 1 ≤ i ≤ r)

the operator H_{b,a} will undergo the perturbation :

δH_{b,a} := \frac{1}{2πi} ∑_{2≤n} ∫_{0}^{∞} H_{b,y} \frac{(a − b) e^{−n (y+c^{2}y^{−1})} v^{n}}{(a − y) (y − b)} δ𝔸_{n} H_{y,a} dy

δh_{a,b} := \frac{1}{2πi} ∑_{2≤n} ∫_{0}^{∞} \frac{(a − b) e^{−n (y+c^{2}y^{−1})} v^{n}}{(a − y) (y − b)} \frac{(h_{y,b}(u))^{n+1}}{∂_{u}h_{y,b}(u)} ∂_{u}h_{a,b}(u) dy

Finally, setting a = z, b = +∞, v = e^{z+c^{2} z^{−1}} we get :

δh_{a,b} := \frac{1}{2πi} ∑_{2≤n} ∫_{0}^{∞} \frac{e^{−n (y+c^{2} y^{−1})} e^{+n (z+c^{2} z^{−1})}u^{n}}{(z − y)(1 + A_{1} Ua_{c}^{1}(y) e^{z+c^{2} z^{−1}} u)^{n+1}}

×\frac{dy}{(1 + A_{1} Ua_{c}^{1}(z) e^{z+c^{2} z^{−1}} u)}  (284)

Reverting to §3.2 and (77), we see that the crucial variable are z (∼ ∞) and w := u e^{z} (∼ 0). More precisely, for canonical synthesis to survive the infinitesimal pertubation we have just performed, the above integral (284) must converge uniformly for z large enough (more precisely : for z in some U-shaped neighbourhood of +∞) and for w small enough (more precisely : in a full neighbourhood of 0).

For a vanishing twist (c = 0), Ua_{c}^{1}(y) reduces to a hyperlogarithmic resurgence monomial and it goes to ∞ as const. log(1/y) when y ↓ +0. So, for z fixed, the denominator 1 + A_{1} Ua_{c}^{1}(y) e^{z+c^{2} z^{−1}} u|_{c=0} ≡ 1 + A_{1} Ua_{0}^{1}(y) w vanishes, when the integration variable y decreases to +0, for at least one value w := w(z, y) which also goes to 0. So the integral (284) cannot be w-analytic at w = 0.

For c > 0 , on the other hand, the paralogarithmic resurgence monomial Ua_{c}^{1}(y) remains bounded on the whole integration axis ℝ^{+} and each n-summand in (284) is indeed w-anlytic at w = 0. ^{111}

11.4 The antipodal pairing.

For c > 0 the twisted monomials are essentially invariant under the involution:

pod :  z ↦ z^{−1} and (ω_{i}, c^{2}ω̄_{i}) ↦ (ω̄_{i}, c^{2}ω_{i})  (285)

More precisely, setting :

pari(M^{ω}) := (−1)^{r(ω)} M^{ω}  (286)

and with SSU_{c}^{•} defined as in §12.3.3, we have :

podU_{c}^{•}(z) × U_{c}^{•}(c^{2} z^{−1}) = SSU_{c}^{•}  (287)

and therefore :

U_{c}^{•}(c^{2} z^{−1}) = pari (^{pod}U_{c}^{•}(z) × SSU_{c}^{•}) = pari (^{pod}U_{c}^{•}(z)) × pari (SSU_{c}^{•}) (288)

Θ_{c,c^{2} z^{−1}} = Iv_{u} . K_{c} . Θ_{c,z} . Iv_{u}  (289)

K_{c} := ∑ pari(SSU_{c}^{•}) 𝔸_{•} = ∑ (−1)^{r(•)} SSU_{c}^{•} 𝔸_{•}  (290)

111As for the n-convergence of the series in (284), it offers nondifficulty as long as we take a admissible pertubation δ𝔸_{n}. But n-convergence is not the real issue here, and indeed we may get rid of it by assuming that only a finite number of δ𝔸_{n} are ≠ 0.

Iv_{u} .φ(u) := − φ(−u) ;  Iv_{u} .u^{n+1}∂_{u} .Iv_{u} = (−1)^{n} u^{n+1} ∂_{u} (291)

So, once again we encounter this strange pairing of the “antipodes”. In fact, any time we perform canonical synthesis, we get two objects “for the price of one” : we simultaneously synthesize one (main) object at z = ∞ , and another at z = 0. These two may, or may not, link up under analytic continuation.

11.5 Iso-invariant deformations.

There exists a closed system of formulae (see §6.7 and [E15]) to describe the exact dependence (partial derivatives, asymptotics, etc) of our canonical resurgence monomials as functions of their variable z, twist c and indices ω_{i}, ω̄_{i}. As a result, one may write down the – often unexpectedly simple – partial differential equations which govern the sundry deformations (isoinvariant, iso-monodromic, iso-resurgent, iso-Galoisian, etc) of our synthesised objects.

Thus, from the c-differentiation rule for the monomials :

\frac{1}{2}c ∂_{c} Ue^{•}(z) = z^{−1} Ue^{•}(z) × (e^{z □ + c^{2} z^{−1} □} UR^{•})  (292)

we get (with the notations of §12.1.3) the c-differentiation rule for the normaliser Θ_{c}:

[ \frac{1}{2}c ∂_{c} , Θ_{c}] = UK . Θ_{c}  (293)
with UK  := ∑ (−1)^{r(•)} (e^{z □ + c^{2} z^{−1} □} UR^{•}) 𝔸_{•}

11.6 Remarks and complements.

Remark 1: Antipodal involution. As already pointed out, our twisted monomials have much the same behaviour at both poles of the Riemann sphere. The exact correspondence has just been described in §6.7 using the so-called antipodal involution (285) :

In terms of the objects being produced, this means that canonical object synthesis automatically generates two objects : the ‘true’ object, local at ∞ and with exactly the prescribed invariants, and a ‘mirror reflection’, local at 0 and with closely related invariants. Depending on the nature of the problem (linear/non-linear, etc) and of the invariants (verification or non-verification of an “overlapping condition”) , these two objects may or may not link up under analytic continuation on the Riemann sphere.

Remark 2: Analogy with q-equations. Authors like Sauloy recently ([S]) observed that q-difference equations are in some sense easier to tackle than difference or differential equations, due to dilations z ↦ q z having two fixed points 0 and ∞, whereas shifts z ↦ a + z have only one, namely ∞. It is certainly no coincidence that the simplest resurgence monomials that permit object synthesis are precisely the twisted ones (c ≠ 0), for whom the antipodal symmetry is restored, whereas the twistless monomials (c = 0), though apparently more simple, turn out to be inadequate for this particular purpose.

Remark 3: Necessity of a one-parameter freedom. The necessity of having at least one degree of freedom in object synthesis has been known since the 1980s at least. This holds even for such elementary objects as linear systems (Example 3). Indeed, in most cases, the twist c must exceed a certain lower bound c_{min} that depends on the invariants {𝔸_{ω}}. However, as already pointed out, there exists an important exception : the so-called unilateral classes, when for instance all non-vanishing 𝔸_{ω} have their indices on the same half-line. There any choice c > 0 will do ! This applies in particular to Example 2 when 𝔸_{−1} = 0

Remark 4: WB derivations and WB monomials : unequal status. Working with well-behaved alien derivations is merely convenient, whereas the recourse to well-behaved resurgence monomials is truly indispensible. There is a subtle difference here, which should be well understood. Indeed, the choice of this or that system of WB alien derivations does not affect the result : it simply gives us a comfortable basis of ALIEN to work with. Besides, there is always the lazy option of working with the lateral alien operators Δ Δ_{ω}^{±}, the only drawback being that the corresponding invariants 𝔸_{ω}^{±} cease to be first-order differential operators. In complete contrast, the synthesised object very much depends on the choice of the system of WB monomials. And in the absence of well-behaved monomials, canonical synthesis would founder altogether.

Let us now collect, in a final section, a number of useful formulas about the twisted monomials and the associated monics.

12 Hyper-, peri-, para-logarithmic monomials and monics: total closure.

12.1 Main objects and notations.

12.1.1 Hyperlogarithms, perilogarithms, paralogarithms.

Perilogarithms have indices ϖ_{i} = (ω_{i}, ω_{i}^{⋆}) ∈ (ℂ^{⋆}, ℂ^{⋆}). Hyperlogarithms have indices ω_{i} ∈ ℂ^{⋆}. Perilogarithms have indices ϖ_{i} = (ω_{i}, ω_{i}^{⋆}) ∈ (ℂ^{⋆}, ℂ^{⋆}) with ω_{i}ω_{i}^{⋆} ∈ ℝ^{+}. Perilogarithms have indices ϖ_{i} = (ω_{i}, c^{2}ω̄_{i}) ∈ (ℂ^{⋆}, ℂ^{⋆}). Usually c is fixed, so that only ω_{i} is mentioned.

12.1.2 ∂- or Δ-friendly monomials and monics.

Monics depend only on the indices ω_{i} or ϖ_{i}. Monomials depend on a variable z as well. ∂-friendly monomials behave simply under ordinary z-differentiation, but less so under alien z-differentiation: their alien derivatives necessarily involve a number of so-called ∂-friendly monics. Δ-friendly monomials carry behave simply under alien z-differentiation, but less so under ordinary z-differentiation: their ordinary derivatives necessarily involve a number of so-called Δ-friendly monics. ∂-friendly (resp Δ-friendly) monomials always carry a calligraphic V^{•} (resp U^{•}) as part of their names while the corresponding monics carry an uppercase V^{•} (resp. U^{•}), sometimes supplemented by a suitable string of pre- or suffixes.

12.1.3 Total closure.

The monomials, as functions of z, are acted upon by one ordinary derivation ∂ := ∂_{z} but by infinitely many independent alien derivations Δ_{ω_{i}} or their variants Δ Δ_{ω} := e^{−ωz}Δ_{ω}. The latter have the advantage of commuting with the ordinary derivation ∂, but at the cost of introducing an exponential factor and thus ceasing to act internally on the ring of formal power series of z^{−1}. To highlight the ∂ ↔ Δ duality, it is sometimes convenient to (formally) regroup all alien derivation into one single symbol:

Δ := ∑ Δ_{ω} ; Δ Δ := ∑ Δ Δ_{ω}

□A^{ϖ} := ∥ω∥ A^{ϖ} := (∑ ω_{i}) A^{ϖ}
□^{⋆}A^{ϖ} := ∥ω^{⋆}∥ A^{ϖ} := (∑ ω_{i}^{⋆}) A^{ϖ}
□_{ω_{i}} := [∂_{ω_{i}}, □]
□_{ω_{i}}^{⋆} := [∂_{ω_{i}^{⋆}}, □^{⋆}]

12.2 Hyperlogarithmic monomials and monics.

12.2.1 Basic hyperlogarithms.

\begin{matrix}Δ-friendly monomials & U^{•}(z) , Ue^{•}(z) & symmetral \\ Δ-friendly monics & U^{•} & alternal \\ Δ-friendly monics & US^{•} , SU^{•} & symmetral \\ ∂-friendly monomials & V^{•}(z) , Ve^{•}(z) & symmetral \\ ∂-friendly monics & V^{•} & alternal \\ ∂-friendly monics & VS^{•} , SV^{•} & symmetral \end{matrix}

12.2.2 Basic relations.

Ue^{•}(z) = exp(z □) . U^{•}(z)
Ve^{•}(z) = exp(z □) . V^{•}(z)
U^{•}(z) = V^{•}(z) ∘ U^{•}
V^{•}(z) = U^{•}(z) ∘ V^{•}
I^{•}  = U^{•} ∘ V^{•}  = V^{•} ∘ U^{•}
1^{•}  = US^{•} × SU^{•}  = VS^{•} × SV^{•}
U^{•}  = US^{•} × I^{•} × SU^{•}  if all ω_{i} ∈ ℝ^{+}
V^{•}  = VS^{•} × I^{•} × SV^{•}  if all ω_{i} ∈ ℝ^{+}

12.2.3 More relations.

∂_{ω_{i}} U^{•}(z) = −U^{•}(z) × (\frac{□_{i}}{□} U^{•}) − z □_{i} U^{•}(z)  (294)
z ∂_{z} U^{•}(z) = −z □ U^{•}(z) − U^{•}(z) × U^{•}  (295)

∂_{ω_{i}} Ue^{•}(z) = −Ue^{•}(z) × (exp(z □).\frac{□_{i}}{□} . U^{•})  (296)
z ∂_{z} Ue^{•}(z) = − Ue^{•}(z) × (exp(z □).U^{•})  (297)

∂_{ω_{i}} U^{•} = + (\frac{□_{i}}{□} U^{•}) × U^{•} − U^{•} × (\frac{□_{i}}{□} U^{•})  (298)
∂_{ω_{i}} US^{•} = + (\frac{□_{i}}{□} U^{•}) × US^{•}  (299)
∂_{ω_{i}} SU^{•} = − U^{•} × (\frac{□_{i}}{□} SU^{•})  (300)

∂_{ω} := ∑ ω_{i}∂_{ω_{i}}

∂_{ω} U^{•} = 0  (301)
∂_{ω} US^{•} = + U^{•} × US^{•}  (302)
∂_{ω} SU^{•} = − SU^{•} × U^{•}  (303)

12.2.4 Ordinary and alien differentiation. The ∂ ↔ Δ duality .

(z ∂_{z} + z □) U^{•}(z) = −U^{•}(z) × U^{•}  (304)
(z ∂_{z} + z □) V^{•}(z) = −V^{•}(z) × I^{•}  (305)
Δ U^{•}(z) = I^{•} × U^{•}(z)  (306)
Δ V^{•}(z) = V^{•} × V^{•}(z)  (307)

with □ standing as usual for multiplication by ∥ • ∥ = ∑ ω_{i} and

I^{ω_{1}} := 1 ;  I^{ω_{1},...,ω_{r}} := 0 if r ≠ 1  (308)

12.3 Perilogarithmic monomials and monics.

12.3.1 Basic perilogarithms. Perilogarithms have indices ϖ := (ϖ_{1}, …, ϖ_{r}) with ϖ_{i} = (ω_{i}, ω_{i}^{⋆}) ∈ (ℂ^{⋆}, ℂ^{⋆}) and a real-positive product : ω_{i} ω_{i}^{⋆} ∈ ℝ^{+}. Antipodal involution # : ^{#}M^{ϖ_{1},...,ϖ_{r}} := M^{ϖ_{r}^{⋆},...,ϖ_{1}^{⋆}} with ϖ^{⋆} = (ω_{i}^{⋆}, ω_{i}) if ϖ = (ω_{i}, ω_{i}^{⋆}) Δ-frienly perilogarithms

primary  secundary  type
monomials  Ua^{•}  U^{•}, Ue^{•}  symmetral
monics  U^{•}  UR^{•}, UL^{•}  alternal
monics  USS^{•}, SSU^{•} US^{•}, SU^{•}  symmetral

D-frienly perilogarithms

primary  secundary  type
monomials  Va^{•}  V^{•}, Ve^{•}  symmetral
monics  V^{•}  VR^{•}, VL^{•}  alternal
monics  VSS^{•}, SSV^{•} VS^{•}, SV^{•}  symmetral

12.3.2 Basic relations.

Ue^{•}(z) = exp( z □ + z^{−1} □^{⋆}). Ua^{•}(z)  (309)
U^{•}(z) = exp( z^{−1} □^{⋆}). Ua^{•}(z)  (310)
1^{•} = USS^{•} × SSU^{•} = US^{•} × SU^{•}  (311)
UR^{•} = USS^{•} × (□^{⋆} SSU^{•})  (312)
UL^{•} = USS^{•} × (□ SSU^{•})  (313)
USS^{•} = US^{•} × ^{#}US^{•}  (314)
SSU^{•} = ^{#}SU^{•} × SU^{•}  (315)
SU^{•} = Ua^{•}(1)  (316)

12.3.3 Integral formulae for the Δ-friendly monomials and monics.

Their main ingredients are the CCI (“Common Core Integrand”) :

CCI := \frac{exp(− ∑ ω_{i}t_{i} − ∑ ω_{i}^{⋆} t_{i}^{−1})}{(t_{r} − t_{r−1}) … (t_{3} − t_{2})(t_{2} − t_{1})}  (317)

and the SPA rule (“Standard Path Averaging”^{112}) for multiple integration.

112see §7

Monomials

Ua^{ϖ}(z)  = SPA ∫_{0}^{∞} CCI (t_{1} − z)^{−1} dt_{1} … dt_{r}
Ue^{ϖ}(z)  = e^{∥ω∥ z+∥ω^{⋆}∥ z^{−1}}  Ua^{ϖ}(z)
U^{ϖ}(z)  =  e^{∥ω^{⋆}∥ z^{−1}}  Ua^{ϖ}(z)

Monics

U^{ϖ}  = SPA ∫_{0}^{∞} CCI (∑ ω_{i}) dt_{1} … dt_{r}
= SPA ∫_{0}^{∞} CCI (∑ ω_{i}^{⋆}/t_{i}^{2}) dt_{1} … dt_{r}
UL^{ϖ}  = SPA ∫_{0}^{∞} CCI (∑ ω_{i}/t_{i}) dt_{1} … dt_{r}
UR^{ϖ}  = SPA ∫_{0}^{∞} CCI (∑ ω_{i}^{⋆}/t_{i}) dt_{1} … dt_{r}
SU^{ϖ}  = SPA ∫_{0}^{∞} CCI (t_{1} − 1)^{−1} dt_{1} … dt_{r}
US^{ϖ}  = SPA ∫_{0}^{∞} CCI (1 − t_{r})^{−1} dt_{1} … dt_{r}
SSU^{ϖ}  = SPA ∫_{0}^{∞} CCI (1/t_{1}) dt_{1} … dy_{r}
USS^{ϖ}  = SPA ∫_{0}^{∞} CCI (−1/t_{r}) dt_{1} … dt_{r}

12.3.4 More relations for the Δ-friendly perilogarithms. Monomials:

∂_{ω_{i}} Ua^{•}(z) = −Ua^{•}(z) × \frac{□_{i}}{□} U^{•} − z □_{i}Ua^{•}(z)  (318)
∂_{ω_{i}^{⋆}} Ua^{•}(z) = +z^{−1} Ua^{•}(z) × USS^{•} × □_{i}^{⋆}SSU^{•} − z^{−1} □_{i}^{⋆} Ua^{•}(z)  (319)
z ∂_{z} Ua^{•}(z) = (−z □ + z^{−1} □^{⋆}) Ua^{•}(z) − Ua^{•}(z) × (U^{•} + z^{−1} UR^{•}) (320)

Monics:

∂_{ω_{i}}U^{•} = +(\frac{□_{i}}{□} U^{•}) × U^{•} − U^{•} × (\frac{□_{i}}{□} U^{•}) − □_{i} UR^{•}  (321)
∂_{ω_{i}^{⋆}}U^{•} = +□ ((□_{i}^{⋆} USS^{•}) × SSU^{•})  (322)
∂_{ω_{i}}USS^{•} = +(\frac{□_{i}}{□} U^{•}) × USS^{•}  (323)
∂_{ω_{i}^{⋆}}USS^{•} = +USS^{•} × (\frac{□_{i}^{⋆}}{□^{⋆}} ^{#}U^{•})  (324)
∂_{ω_{i}} SSU^{•} = −SSU^{•} × (\frac{□_{i}}{□} U^{•})  (325)
∂_{ω_{i}^{⋆}} SSU^{•} = −(\frac{□_{i}^{⋆}}{□^{⋆}} ^{#}U^{•}) × SSU^{•}  (326)

∂_{ω_{i}} US^{•} = +(\frac{□_{i}}{□} U^{•}) × US^{•} − □_{i} US^{•}  (327)
∂_{ω_{i}^{⋆}} US^{•} = +US^{•} × (□_{i}^{⋆} ^{#}US^{•}) × SU^{•}  (328)
∂_{ω_{i}} ^{#}US^{•} = +SU^{•} × (□_{i} US^{•}) × ^{#}US^{•}  (329)
∂_{ω_{i}^{⋆}} ^{#}US^{•} = +^{#}US^{•} × (\frac{□_{i}^{⋆}}{□^{⋆}} ^{#}U^{•}) − □_{i}^{⋆} ^{#}US^{•}  (330)
∂_{ω_{i}} SU^{•} = −SU^{•} × (\frac{□_{i}}{□} U^{•}) − □_{i} SU^{•}  (331)
∂_{ω_{i}^{⋆}} SU^{•} = +^{#}US^{•} × (□_{i}^{⋆} ^{#}SU^{•}) × SU^{•}  (332)
∂_{ω_{i}} ^{#}SU^{•} = +^{#}SU^{•} × (□_{i} SU^{•}) × US^{•}  (333)
∂_{ω_{i}^{⋆}} ^{#}SU^{•} = −(\frac{□_{i}^{⋆}}{□^{⋆}} ^{#}U^{•}) × ^{#}SU^{•} − □_{i}^{⋆} ^{#}SU^{•}  (334)

12.3.5 Yet more relations for the Δ-friendly perilogarithms. The partial differentiation rules relative to

∂_{ω} := ∑ ω_{i}∂_{ω_{i}} and ∂_{ω^{⋆}} := ∑ ω_{i}^{⋆}∂_{ω_{i}^{⋆}}

though deducible from the above, are also worth mentioning. They become particularly useful in the paralogarithmic case, since

∂_{ω} ≡ ∂_{ω^{⋆}} ≡ 1/2 c ∂_{c} ≡ c^{2} ∂_{c^{2}}

∂_{ω} Ua^{•}(z) = − Ua^{•}(z) × U^{•} − z □ Ua^{•}(z)  (335)
∂_{ω^{⋆}} Ua^{•}(z) = + z^{−1} Ua^{•}(z) × UR^{•} − z^{−1} □ Ua^{•}(z)  (336)

\begin{aligned}∂_{ω} U^{•} & = & ∂_{ω^{⋆}} U^{•} & = & □ ((□^{⋆} USS^{•}) × SSU^{•} & = & −□UR^{•} \\ ∂_{ω} USS^{•} & = & ∂_{ω^{⋆}} USS^{•} & = & +U^{•} × USS^{•} & = & USS^{•} ×^{#}U^{•} \\ ∂_{ω} SSU^{•} & = & ∂_{ω^{⋆}} SSU^{•} & = & −SSU^{•} × U^{•} & = & −^{#}U^{•}SSU^{•} \end{aligned}

∂_{ω} US^{•} = +U^{•} × US^{•} − □ US^{•}  (337)
∂_{ω^{⋆}} US^{•} = +US^{•} × (□^{⋆} ^{#}US^{•}) ×^{#}SU^{•}  (338)
∂_{ω} SU^{•} = −SU^{•} × U^{•} − □ SU^{•}  (339)
∂_{ω^{⋆}} SU^{•} = +^{#}US^{•} × (□^{⋆} ^{#}SU^{•}) × SU^{•}  (340)

12.3.6 A glimpse of the ∂-friendly perilogarithms.

(z ∂_{z} + z □ − z^{−1} □^{⋆}) Va^{•}(z) = −Va^{•}(z) × Ja^{•}(z)  (341)

with an elementary, one-component mould Ja^{•}:

Ja^{\underline{ϖ}^{1}}(z) = Ja^{( {ϖ_{1} \atop τ_{1}} )}(z) := 1 if τ_{1} = 0
:= cz^{−1} if τ_{1} = 1
Ja^{\underline{ϖ}_{1},...,\underline{ϖ}_{r}}(z) = Ja^{( {ϖ_{1} \atop τ_{1}} {,..., \atop ,...,} {ϖr \atop τr})}(z) := 0 if r ≠ 1

and with discrete indices τ_{i} ∈ {0, 1}.

12.3.7 From ∂- to Δ-friendly .

Ua^{•}(z) = Va^{•} ∘ U^{•}  (342)

which is short-hand for

Ua^{ϖ}(z) = ∑_{\{ {ϖ^{1}...ϖ^{s}=ϖ \atop s≥1 , τ_{i}∈\{0,1\}} \}} Va^{( {∥ϖ^{1}∥ \atop τ_{1}} {,..., \atop ,...,} {∥ϖ^{s}∥ \atop τs})}(z) U^{( {ϖ^{1} \atop τ_{1}} )} … U^{( {ϖ^{s} \atop τs} )}  (343)

with

U^{\underline{ϖ}} = U^{( {ϖ \atop τ_{0}} )} := U^{ϖ} if τ_{0} = 0
:= UR^{ϖ} if τ_{0} = 1

12.3.8 Resurgence equations.

Δ U^{•} = I^{•} × U^{•}  ( with indices ϖ_{i})  (344)
Δ V^{•} = V^{•} × V^{•} ( with indices \underline{ϖ}_{i} = ({ϖ_{i} \atop τ_{i}}))  (345)

The endearingly simple relation I^{•} = V^{•} ∘ U^{•} connecting the Δ- and ∂- friendly hyperlogarithmic monics carries over to the perilogarithmic monics, but with doubled storeyed indices \underline{ϖ}_{i} = ({ϖ_{i} \atop τ_{i}}) in the moulds and a doublestoreyed mould composition ∘ interpreted as above.

12.4 Paralogarithmic monomials and monics.

We now replace the antipodal involution :

(^{#}M)^{ϖ_{1},...,ϖ_{r}} := M^{c^{2}ϖ̄_{r},...,c^{2}ϖ̄_{1}}  (346)

by the more convenient variant :

(^{♯}M)^{ϖ_{1},...,ϖ_{r}} := M^{ϖ̄_{r},...,ϖ̄_{1}}  (347)

and we get :

♯SSU_{c}^{•} = SSU_{c}^{•}  (348)
♯USS_{c}^{•} = USS_{c}^{•}  (349)
♯U_{c}^{•} = SSU_{c}^{•} × U_{c}^{•} × USS_{c}^{•}  (350)
♯UR_{c}^{•} = SSU_{c}^{•} × UL_{c}^{•} × USS_{c}^{•}  (351)
♯UL_{c}^{•} = SSU_{c}^{•} × UR_{c}^{•} × USS_{c}^{•}  (352)

(^{♯}Ua)_{c}^{•}(z) × (Ua)_{c}^{•}(c^{2}/z) ≡ SSU_{c}^{•}  (353)

The integral formulae of §12.3.3 remain unchanged, except that the extreme factors (y_{1} − 1)^{−1} and (1 − y_{r})^{−1} become (y_{1} − c)^{−1} and (c − y_{r})^{−1}. One should always integrate along the axes arg(ω_{i} y_{i}) = arg(ω̄_{i}/y_{i}) = 0 and heed the “SPA” rules of mutual bypassing whenever several consecutive Arg(ω_{i}) coincide. The partial differentiation rules for the perilogarithms particularise to the paralogarithms.

PS. I wish to thank David Sauzin for checking the formulae of §12.

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