Intertwined mappings

Jean Ecalle (Orsay)  Bruno Vallet (Orsay)

November 22, 2003

Abstract : We show that, contrary to expectations, there exist pairs of formal and even analytic, non-commuting and non-elementary (neither algebraic nor algebraic-differential) mapping germs in Diff(ℂ, 0) that are ‘entwined’ in a group relation W (f, g) = id. In the case of identity-tangent mappings, ‘twins’ exhibit, rather than analyticity, generic divergence , but of a particularly interesting sort : resurgent, accelero-summable, and with simple alien derivatives.

Contents

1 Introduction. Intertwined mappings and bound groups. 1 1.1 Interwined mappings and bound groups. . . . . . . . . . . . . 1 1.2 Motivations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Reminders about local diffeos of ℂ. . . . . . . . . . . . . . . . 3 1.4 Free groups and algebras. . . . . . . . . . . . . . . . . . . . . 6 1.5 Campbell-Hausdorff type formulae. . . . . . . . . . . . . . . . 6 1.6 General notations. . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7 Outline of the paper. . . . . . . . . . . . . . . . . . . . . . . . 7

2 Some group theory. Alternators and word factorisation. Periodic automorphisms. 9 2.1 Filtrations and graduations on Gr and Lie. . . . . . . . . . . 9 2.2 Finite criteria for alternance (d_{1}, d_{2}). . . . . . . . . . . . . . . 12 2.3 Low-complexity alternators. Measures of word complexity… 13 2.4 Periodic automorphisms of Gr . . . . . . . . . . . . . . . . . . 14 3 Identity-tangent twins: twenty typical examples. 16

3.1 Collapsors and divisors . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Normal-conormal form of twins. Connectors. . . . . . . . . . . 18 3.3 Main types of twin-begetting relators W (A, B). . . . . . . . 20 3.4 Basic examples. . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 Examples with built-in symmetries . . . . . . . . . . . . . . . 33 3.6 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Some Lie theory. Active/passive subalgebras. Divisors and universal kernels. 45 4.1 The active/passive filtration of Lie[a, b]. . . . . . . . . . . . . 45 4.2 Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.3 Universal constraints. . . . . . . . . . . . . . . . . . . . . . . . 52 4.4 Lagrangian constraints. . . . . . . . . . . . . . . . . . . . . . . 53 4.5 Power constraints. . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.6 Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Generic, low-complexity identity-tangent twins. 59 5.1 Finite co-dimension. The general picture. . . . . . . . . . . . . 59 5.2 Fixed ratio p/q and no continuous parameter. . . . . . . . . . 60 5.3 Fixed ratio p/q and one continuous parameter. . . . . . . . . . 61 5.4 Free ratio p/q and one continuous parameter. . . . . . . . . . 62

6 Non identity-tangent twins. 64 6.1 Twins of tangency (0, q). . . . . . . . . . . . . . . . . . . . . . 64 6.2 Simple examples. . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.3 Twins of tangency (0, 0). . . . . . . . . . . . . . . . . . . . . . 67 6.4 Simple examples. . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.5 Pre-identity-tangent ‘twins’: type (0^{∗}, 0^{∗}). . . . . . . . . . . . 67 6.6 Simple examples. . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.7 Siblings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.8 Simple examples. . . . . . . . . . . . . . . . . . . . . . . . . . 70

7 Analytic nature of twins. 71 7.1 Multiplier |a| ≠ 1: convergence. . . . . . . . . . . . . . . . . . 71 7.2 Multiplier |a| = 1: divergence. . . . . . . . . . . . . . . . . . . 73 7.3 Pre-identity-tangent ‘twins’: convergence. . . . . . . . . . . . . 74 7.4 Identity-tangent twins : the general picture. . . . . . . . . . . 74 7.5 Identity-tangent twins : autonomous differential-difference equa-

tions and double resurgence. . . . . . . . . . . . . . . . . . . . 75 7.6 Identity-tangent twins : canonical accelero-summability. . . . . 79 7.7 Identity-tangent twins : ‘shadow twins’ and obstructions to

analyticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8 Transserial twins. 81 8.1 Reminders about transseries. . . . . . . . . . . . . . . . . . . . 81 8.2 Transserial twins of exponential or transexponential type. … 83 8.3 Van der Hoeven’s intermediate value theorem. . . . . . . . . . 84 8.4 Canonical resummation of transserial twins. . . . . . . . . . . 84 8.5 The ordering of free or bound groups. Natural and exotic

orders. Ultraexponential arbitration. . . . . . . . . . . . . . . 86

9 Conclusion : settled points, open questions. 88

1 Introduction. Intertwined mappings and bound groups.

1.1 Interwined mappings and bound groups.

The mappings we shall be concerned with are mainly the local diffeomorphisms (or “diffeos” for short) of ℂ (with either x = 0 or z = ∞ as fixed point) as well as their formal counterparts. More to the point, we shall be looking for “intertwined mappings”, i.e. for pairs (f, g) of diffeos (or “twins”) that generate a ‘bound’ group G{f, g}, by which we mean a group that is

(i) non-free, i.e not isomorphic to the two-generator free group

(ii) non-abelian

(iii) non-elementary, i.e. not conjugate to a group of homographies.

The last requirement is there to rule out uninteresting situations. It means that no local change of variable, whether analytic or formal, unramified ( x ⟼ cx + ⋯; z ⟼ a z + ⋯) or ramified ( x ⟼ c x^{n} + ⋯; z ⟼ a z^{n} +⋯) should turn f and g simultaneously into a pair of “homographies” or “möbius maps” x ⟼ ax(1 + b x)^{−1} (near x = 0) or, if the fixed point be z = ∞, into a pair of “similitudes” z ⟼ a z + b.

In order to generate a bound group in the above sense, f and g must of course verify some composition relation :

W (f, g) = id  with  (1.1)
W (f, g) = f^{∘m_{1}} ∘ g^{∘n_{1}} ∘ ⋯ f^{∘m_{r}} ∘ g^{∘n_{r}}  (m_{i}, n_{i} ∈ ℤ^{∗}; r ∈ ℕ^{∗})

The point, however, is that most relations W force f and g to commute or “nearly-commute”, i.e. to be simultaneously conjugate to homographies. In fact, as we shall see, only finely honed and fairly intricate relations W can lead to suitable solutions (f, g) and give rise to bound groups G{f, g}.

1.2 Motivations.

(i) Originally, the question as to the existence of bound groups (specifically, for analytic, identity-tangent local self-mappings of ℂ ) was raised in the early 90s ago by D. Cerveau, R. Moussu and others in connection with the classification of local analytic foliations on ℂ^{2} and the holonomy of local differential equations (see [Ce]). The expectation, it seems, was that such groups didn’t exist. In 1995, however, one of us came up with a first series of examples, namely those listed in the mid-part of §3 (mainly §3.3) . But then the subject was provisionally laid to rest, and only recently taken up anew for systematic investigation

(ii) Another way of looking at the same question ([Ce]) is by trying to extend the so-called Tits alternative. In its classical formulation, the Tits alternative applies to subgroups G ⊆ Gl(n, ℂ) of the linear group, and it states that

– either G is virtually solvable, ie it contains a finite solvable subgroup of finite index

– or it contains a free 2-generator subgroup, which is Zariski dense. So a natural and important question is : does a similar alternative hold for Diff(ℂ, 0) and its formal counterpart ? For the latter, we shall show that the answer is no. For the former also, the answer is no, but the identity-tangent sub-case is still open.

(iii) Actually, bound groups of mappings are basic and natural objects in their own right and their study requires no elaborate apology. Indeed, in the one-operation-only context of group theory, “being intertwined ” is for a pair of mappings (f, g) the closest equivalent one could think of to “algebraic dependence ” for a pair of irrational numbers (a, b). Moreover, since twins (f, g) are so scarce, so thinly spread out in the landscape, their precise shape and properties are a matter of legitimate curiosity.

(iv) The search for twins (and more generally for siblings, ie “overrelated” systems of diffeomorphisms; see §6.7-8) is closely bound up with the topic of infinite-dimensional group representations (“into”) and realisations (“onto”). Whereas finite-dimensional representations and realisations are quite adequate for finite groups, infinite groups (especially solvable ones) often call for infinite-dimensional representations, and in particuliar for realisations as special groups of mappings (local or global).

(v) Twins provide a striking illustration of the ubiquity of resurgence and the usefulness of acceleration. Indeed, as we shall show, although identitytangent twins are generically (and probably always) divergent, they are always resurgent (in a suitable chart) and even biresurgent, i.e. resurgent with respect to two distinct “critical times”. Therefore, to resum them, we must resort not only to the usual Borel-Laplace scheme, but also to one intermediary acceleration transform.

1.3 Reminders about local diffeos of ℂ.

Rather than working directly with local diffeos f :

f : x ⟼ ax{1 + ∑_{n≥1} a_{n}x^{n}} (x ∼ 0; a ≠ 0)  (1.2)

we shall give precedence to the associated substitution operators F, denoted by the same letters, but capitalised :

F : φ ⟼ F.φ := F ∘ φ (∀φ ∈ ℂ[[x]])  (1.3)

The reason is that operators afford greater flexibility : instead of one single operation for diffeos (namely the composition product ∘), operators can be subjected to four operations: addition; multiplication; the taking of the logarithm (or infinitesimal generator) and the Lie bracketing of infinitesimal generators. But one should be mindful of the order reversal: the operator product F G corresponds to the composition product g ∘ f.

Diffeos (1.2) with a “multiplier” a = 1 are said to be identity-tangent. When a is ≠ 1 but a unit root, f is said to be pre-identity tangent (since a suitable iterate is identity tangent). When neither is the case, f is declared non-identity tangent.

If f is identity-tangent, the substitution operator F is triangular with respect to the natural basis {x^{n}} of ℂ[[x]]. Moreover, F −1 is triangular with a zero-diagonal, and so is F_{∗} := logF, but whereas F is an automorphism of the algebra C[[x]]:

F (φ.ψ) ≡ F (φ).F (ψ) (∀φ, ψ ∈ C[[x]])  (1.4)

its logarithm F_{∗} is a derivation and thus of the form:

F_{∗} = f_{∗}(x) ∂ with f_{∗}(x) = ∑_{n≥1} a_{∗n} x^{n+1} and ∂ = ∂_{x}.  (1.5)

The tangency order p (≥ 2) of an identity-tangent diffeo f is the index of the first non zero coefficient a_{n} in (1.2) or a_{∗n} in (1.5), which amounts to the same, since the leading coefficients coincide in both series (a_{p} = a_{∗p}).

If f has tangency order p, then under some (formal, unramified) change of coordinate it can be brought to the normal form :

f^{nor}(x) ≡ x \{1 − \frac{1}{p} x^{p} + (\frac{1}{2} \frac{1 + p}{p^{2}} − \frac{α_{∗}}{p}) x^{2p} + ⋯ \}  (1.6)

F^{nor} ≡ exp(F_{∗}^{nor}) ≡ postcomposition by f^{nor}  (1.7)

F_{∗}^{nor} ≡ −\frac{1}{p} x^{p+1} (1 + α_{∗} x^{p})^{−1} ∂  (1.8)

with a well-defined iteration residue α_{∗} ∈ ℂ. As a consequence, a further (ramified) change of coordinate :

x ⟼ z := x^{−p} + α_{∗} log(x^{−p})  (1.9)

will turn f into the plain unit shift z ⟼ z + 1 with the fixed point moving from x = 0 to z = ∞.

If f is pre-identity-tangent, its multiplier has a primitive unit root a = exp(2 π \frac{p_{∗∗}}{p_{∗}}) of some order p_{∗}. So the iterate f^{p ∗} is identity-tangent, and f itself can be normalised to:

F^{nor} = R . exp(−\frac{1}{p} x^{p+1} (1 + α_{∗} x^{p})^{−1}) ∂)  (1.10)

with a well-defined iteration residue α_{∗}, a tangency order p divisible by p_{∗}, and with R denoting the rotation operator of angle −2 π \frac{p_{∗}}{p_{∗∗}} :

R . φ := φ (x . exp(−2 π i \frac{p_{∗}}{p_{∗∗}}))  (1.11)

Lastly, if f is non-identity-tangent, it can be linearised, at least formally, i.e. normalised to f^{nor}(x) = a x.

Throughout the first half of this investigation (right through §5), we shall be concerned with formal twins, without any growth restriction on the coefficients a_{n} in (1.2). Only in the second half shall we look for analytic twins or, failing that, for resurgent ones. But right now, we must briefly mention how analyticity affects the main constructs pertaining to diffeos.

For non-identity-tangent analytic diffeos f, the normalising (i.e. linearising) change of coordinate (unique upto a linear map x ⟼ c x) is always analytic, except if the multiplier a of f is of module 1 and Liouvillian, which in this context means that it violates A.D. Bryuno’s diophantine condition. In that case, the normalising map is generically divergent and non-resummable.

For identity-tangent (or pre-identity-tangent) analytic diffeos, on the other hand, both the normalising change of coordinate and the infinitesimal generator f_{∗} are (generically) divergent but (always) resurgent (with critical time z = x^{−p}) and Borel-Laplace resummable. Such diffeos, on top of their two formal invariants (the tangency order p and the iteration residue α_{∗}) possess a countable infinity of independent analytic invariants, each of which depends holomorphically on f (see [E2]).

All these results carry over, with only slight changes, to the context of higher-dimensional local diffeos (i.e. diffeos of ℂ_{0}^{ν}), but there is one respect in which one-dimensional diffeos stand out : for them, the set of all iterates f^{w} of complex order w always coincides with the centraliser of f, that is, with the set of all diffeos g that commute with f.

1.4 Free groups and algebras.

Let Al(a, b) (resp. Lie(a, b) denote the associative (resp. Lie) algebra freely generated by two non-commuting symbols a and b, with the natural immersion Lie(a, b) ⊂ Al(a, b). Further, let Gr(A, B) be the free non-abelian group generated by A and B. By substituting (ea, eb) for (A,B), we may regard Gr(A, B) as a multiplicative subset of \overline{Al}(a, b) and log(Gr(A, B)) as a subset of \overline{Lie}(a, b), where \overline{Al} and \overline{Lie} denote the natural completion of Al and Lie obtained by allowing infinite sums.

A polynomial function on Gr(A, B) is a (scalar-valued) function P of the form :

P (A^{m_{1}} B^{n_{1}} … A^{m_{r}} B^{n_{r}}) ≡ P (r ∥ m_{1}, n_{1}, …, m_{r}, n_{r})  (1.12)
with r ∈ ℕ ; m_{i}, n_{i} ∈ ℤ)

with polynomial dependence on the variables m_{i} and n_{i} and the natural connection condition :

P (r ∥ …, m_{j}, 0, m_{j+1}, …) ≡ P (r − 1 ∥ …, m_{j} + m_{j+1}, …) (1.13)
P (r ∥ …, n_{j}, 0, n_{j+1}, …) ≡ P (r − 1 ∥ …, n_{j} + n_{j+1}, …)  (1.14)

A subset H of Gr(A, B) is said to be of (polynomial) codimension d if it is the zero locus of d independent polynomial functions on Gr(A, B) :

H = {W ∈ Gr(A, B) ; P_{1}(W) = P_{2}(W) = ⋯ P_{d}(W) = 0} (1.15)

If, for each monomial c = a^{μ_{1}} b^{ν_{1}} ⋯ a^{μ_{s}} b^{ν_{s}} in Al(a, b) and each element W (A, B) = A^{m_{1}} B^{n_{1}} ⋯ A^{m_{s}} B^{n_{s}} in Gr(A, B), we define P_{c}(W) as the (rational) coefficient of c in the natural expansion of W (e^{a}, e^{b}) then the aggregate of all these P_{c} clearly constitutes a complete but non-free set of polynomial functions on Gr(A, B). But if we let γ run through some basis (eg Lyndon basis) of Lie(a, b) and define Pγ^{∗}(W) as the coefficient of log(W (e^{a}, e^{b}) along the basis vector γ, then these Pγ^{∗} yield a set of polynomial functions on Gr(A, B) that is still complete but free as well.

1.5 Campbell-Hausdorff type formulae.

We shall make constant use of the notations :

\overline{b} a := [b , a] := b a − a b  (1.16)

\overline{\overline{B}} A := \{B , A\} := B^{−1} A^{−1} B A  (1.17)

We shall also require the Campbell-Hausdorff formula :

log(eb ea) = b + a + \frac{1}{2} [b, a] + \frac{1}{12} [b, [b, a]] + \frac{1}{12} [a, [a, b]] + … (1.18)

along with these two variants:

log \{eb, ea\} := log(e^{−b} e^{−a} eb ea) = [b, a] − \frac{1}{2} [b + a, [b, a]] + … (1.19)

log {ea^{r}, {ea^{r−1}⋯ {ea^{2}, ea^{1}} ⋯ }}
= \overline{a}_{r} \overline{a}_{r−1}⋯ \overline{a}_{2} a_{1} − \frac{1}{2} ∑_{j=1}^{r} \overline{a}_{r} \overline{a}_{r−1} ⋯ \overline{a}_{j}^{2} ⋯ \overline{a}_{2} a_{1} + … (1.20)

where the first term in the sum ∑ (corresponding to j = 1) has to be interpreted as \overline{a}_{r} ⋯ \overline{a}_{3} \overline{a}_{1} \overline{a}_{2} a_{1} or, equivalently, as −\overline{a}_{r} ⋯ \overline{a}_{3} \overline{a}_{1}^{2} a_{2}.

1.6 General notations.

Boldface symbols shall be reserved almost exclusively for free structures (groups, associative algebras, Lie algebras) and their elements. Plain symbols shall be used for bound structures, such as the group G̃ (resp. G) of all formal (resp. analytic) local diffeomorphisms of ℂ. As a rule, the twiddle ∼ shall signal formalness (as opposed to analyticity) in series, diffeos, differential equations etc ... But in practice, to avoid unnecessary clumsiness, the twiddle will mark only sets (such as G̃) and seldom their elements (f,g etc ...), except when strictly indispensible, for instance whenever we must carefully distinguish between a divergent formal power series φ̃ and its (not necessarily unique) sums φ obtained by various resummation procedures.

As already pointed out, diffeomorphisms shall be denoted by small letters f,g, etc, while the corresponding substitution operators F,G,etc, shall always be capitalised. For expediency, both will usually be referred to as “diffeos”.

1.7 Outline of the paper.

The paper’s first half is devoted to the construction of formal twins. After some group-theoretic reminders (§2), we go through a series of easy-to-construct examples (§3) illustrative of the main types of (identity-tangent) twins. Then, after a Lie-theoretic excursion with some novel material (§4), we attempt a reasonable systematization (§§5, 6), as opposed to an exhaustive one which, as we shall see, is completely out of reach. For each of the leading twin types, we carefully indicate the simplest defining relations W, i.e. those with lowest (polynomial) codimension or with other low “complexity indices”. We then sketch a natural generalisation : bound groups with more than two generators (“siblings”.)^{1}

In the next section (§7) we investigate the analytic nature of the twins constructed thus far. Since the solutions (F, G) of any equation W (F, G) = 1 are defined upto a simultaneous conjugacy:

(F, G) ⟼ (H^{−1} F H, H^{−1} G H)  (1.21)

the pertinent question is not, of course, whether the general solution converges, but rather: how simple can the pair (F, G) be made in a suitable chart ? Or again : how simple does F (resp. G) become after its twin G (resp. F) has been normalised ? For non-identity tangent twins, it is rather easy (by applying the theory of the so-called “sandwich equation”) to show the generic analycity of well-chosen representatives (F, G). For identity-tangent twins, the position is exactly the reverse: we establish their generic divergence and resurgence. It would even seem that identity-tangent twins are always divergent and always resurgent, but a regular proof (especially of the former) appears to be a long way off.

The last, very sketchy section (§8) discusses twins in the larger setting of transseries and analysable functions (as opposed to power series and analytic or resurgent germs) and broaches a number of side-issues, such as the ordering of free groups. ^{2}

1A section devoted to yet another generalisation – bound groups consisting of higherdimensional diffeos – was scrapped for lack of space.

2another such side-issue, originally meant for inclusion in this paper, has been removed for lack of space and redirected to [EV2] . It asks : what is the “proper” and most “comprehensive” notion of analyticity on free algebras (Lie or associative) ? We rephrase the question so as to give it a clear-cut meaning. Then we solve it and come up with a startling, quite counter-intuitive answer.

2 Some group theory. Alternators and word factorisation. Periodic automorphisms.

Throughout this section, Gr and Lie will stand for Gr(A, B) and Lie(a, b), i.e. will denote the two-generator free group resp. Lie algebra.

2.1 Filtrations and graduations on Gr and Lie.

The algebra Lie has an elementary graduation into homogeneous components

Lie = ⨁_{d≥1} Lie^{[d]} = ⨁_{d_{i}≥1} Lie^{[d_{1},d_{2}]}  (2.1)

with Lie^{[d]} (resp. Lie^{[d_{1},d_{2}]}) spanned by multicommutators of global degree d (resp. of degree d_{1} and d_{2} in a and b). The corresponding dimensions are given by classical formulas:

lie(d) := dim Lie^{[d]} := \frac{1}{d} ∑_{δ|d} μ(δ) 2^{d/δ}  (2.2)

lie(d_{1}, d_{2}) := dim Lie^{[d_{1},d_{2}]} := \frac{1}{d_{1} + d_{2}} ∑_{δ|d_{i}} \frac{((d_{1} + d_{2})/δ) !}{(d_{1}/δ) ! (d_{2}/δ) !}  (2.3)

μ(•) being the Möbius function. Going with that graduation we have the filtration:

Lie = ⋃_{d} Lie^{{d}} = ⋃_{d_{i}} Lie^{{d_{1},d_{2}}}  (2.4)

Lie^{{d}} = ⨁_{d′≥d} Lie^{[d′]} ; Lie^{{d_{1},d_{2}}} = ⨁_{d′_{i}≥d_{i}} Lie^{[d′_{1},d′_{2}]}  (2.5)

with the obvious commutation inclusions.

As far as the algebra Lie is concerned, the graduation is the primary notion and the filtration rather derivative. But with the group Gr the position is exactly the reverse. Moreover, we seem to have the choice between two alternative, a priori non-equivalent definitions for the filtration on Gr. According to the first definition, Gr^{{d}} (resp. Gr^{{d_{1},d_{2}}}) is simply the set of all W (A, B) in Gr whose natural Lie image

w(a, b) := log W (e^{a}, e^{b})  (2.6)

lies in Lie^{{d}} (resp. Lie^{{d_{1},d_{2}}}). This defines a filtration, i.e.

{Gr^{{d′}}, Gr^{{d′′}}} ⊂ Gr^{{d′+d′′}}  (2.7)
; {Gr^{{d′_{1},d′_{2}}}, Gr^{{d′′_{1},d′′_{2}}}} ⊂ Gr^{{d′_{1}+d′′_{1},d′_{2}+d′′_{2}}}

because the Campbell-Hausdorff formula (1.19) for brackets involves, on its right-hand side, only terms of degree ≥ 1 in a and b.

Then we have a second definition (provisionnally distinguished by double braces) according to which Gr^{{{d}}} (resp. Gr^{{{d_{1},d_{2}}}}) is the subgroup generated by all multicommutators of alternance ≥ d (resp. ≥ (d_{1}, d_{2})). We speak here of “alternance” rather than “degree” to preclude any confusion with the “degree” of a word W (A, B). A multicommutator of alternance d is of course one with exactly d arguments (which we may take to be A^{±1} or B^{±1}) and a multicommutator of alternance (d_{1}, d_{2}) is one with exactly d_{1} arguments A^{m_{i}} (or A^{±1}) and d_{2} arguments B^{n_{i}} (or B^{±1}). The reason why we may assume all exponents m_{i} and n_{i} to be ±1 is that by repeated use of the Witt-Hall identities:

{A, B} {B, A} = 1  (2.8)
{A, B C} = {A, C} {A, B} {{A, B}, C}  (2.9)
{A B, C} = {A, C} {{A, C}, B}{B, C}  (2.10)

we may break up any multicommutator into a product of multicommutators with equal (or greater) alternance, but with arguments of the form A^{±1} or B^{±1}.

Clearly, the Gr^{{{d}}} and Gr^{{{d_{1},d_{2}}}} define a new filtration on Gr, i.e. the inclusions (2.7) extend to the double-braced Gr^{{{··· }}}. But in fact:

Proposition 2.1 The two natural filtrations on Gr coincide:

Gr^{{d}} = Gr^{{{d}}}  (2.11)
Gr^{{d_{1},d_{2}}} = Gr^{{{d_{1},d_{2}}}}  (2.12)

Corollary 2.1 The quotients:

Gr^{[d]} := Gr^{{d}}/Gr^{{d+1}}  (2.13)
Gr^{[d_{1},d_{2}]} := Gr^{{{d_{1},d_{2}}}}/(Gr^{{{d_{1}+1,d_{2}}}} . Gr^{{{d_{1},d_{2}+1}}})  (2.14)

make sense (since the groups right of the slash are distinguished subgroups of those left of the slash) and define abelian groups Gr^{[d]} and Gr^{[d_{1},d_{2}]} that are isomorphic to Lie^{[d]} and Lie^{[d_{1},d_{2}]} (or rather to their additive submodules spanned by multicommutators).

Proof : The inclusions Gr^{{··· }} ⊃ Gr^{{{··· }}} are trivial, since formula (1.20) carries on its right-hand side only multicommutators of degree ≥ 1 in each of the variables a_{i}. The reverse inclusions, however, are by no means obvious. The one involving the “global” filtration, namely Gr^{{d}} ⊂ Gr^{{{d}}}, is a classical result by P. Hall (see [L.S]),which involves much more than dabbling with the Witt-Hall identities (2.8), (2.9), (2.10). Fortunately, it is but a short step from this inclusion to the more precise inclusion Gr^{{d_{1},d_{2}}} ⊂ Gr^{{{d_{1},d_{2}}}}. In fact, all it takes is proving the following :

Lemma 2.1 Word factorisation :

For each W (A, B) in Gr^{{d}} (d ≥ 1) and each pair (d_{1}(•), d_{2}(•)) of ‘complementary’ permutations of {1, …, d − 1} (i.e. permutations such that d_{1}(i) + d_{2}(i) ≡ d), there exists a (non-unique) factorisation:

W (A, B) = W_{1}(A, B) W_{2}(A, B) ⋯ W_{r−1}(A, B)  (2.15)
with W_{i} ∈ Gr^{{{d_{1}(i),d_{2}(i)}}}

The argument being much the same for all permutations, let us check this for d_{1}(i) ≡ i. Due to the inclusion Gr^{{d}} ⊂ Gr^{{{d}}} and by repeated use of the Witt-Hall identities, we can produce a factorisation W = V_{1} V_{2} ⋯ V_{s} with elementary factors V_{j} of alternance (d′_{j}, d′′_{j}) in (A, B) with d′_{j} + d′′_{j} ≥ d. Set δ = inf d′_{j} and write W = P Q V_{j_{1}} R, with P = V_{1} V_{2} ⋯ V_{j_{1}} regrouping all factors V_{j} (if any) with alternance δ in A and already in front position; with V_{j_{1}} denoting the first factor (if any) with A-alternance δ but not in front position; and with Q regrouping all remaining factors. This leads to a new factorisation:

W = P V_{j_{1}} {V_{j_{1}}, Q^{−1}} Q R  (2.16)

with a factor {V_{j_{1}}, Q^{−1}} Q which in turn may be broken up into elementary factors V′_{j} of A-alternance > 2 δ. By repeating the process for the remaining factors V_{j_{1}}, V_{j_{2}}, etc ... (if there be such) of A-alternance δ in Q, we can move all these factors into front position; and the all factors with A-alternance δ+1, δ + 2, etc ...

This proves the lemma, which we may now apply to words W (A, B) in Gr^{{d_{1},d_{2}}}, in which case the decomposition (2.15) will reduce to only one non-trivial factor, that single factor being in Gr^{{{d_{1},d_{2}}}}. So Gr^{{d_{1},d_{2}}} ⊂ Gr^{{{d_{1},d_{2}}}}, which completes the proof of Proposition 2.1. The corollary about the structure of Gr^{[d]} and Gr^{[d_{1},d_{2}]} is then an easy consequence.

2.2 Finite criteria for alternance (d_{1}, d_{2}).

Each subgroup Gr^{{d}} has finite codimension in Gr (see §1.4). In fact, its codimension is exactly lie(2) + ⋯ + lie(d − 1). But, save for (d_{1}, d_{2}) = (1, 1), each subgroup Gr^{{d_{1},d_{2}}} has infinite codimension. Indeed, if we go by the (first) definition of Gr^{{d_{1},d_{2}}}, checking that a given word W (A, B) is in Gr^{{d_{1},d_{2}}} implies checking infinitely many “polynomial” identities P (W) = 0 of type (1.12).

Fortunately, there exist criteria involving a finite number of steps. First, observe that any W (A, B) in Gr^{{1,1}} may be written in a unique way as finite products of factors A_{n} := B^{n} A B^{−n} or B_{m} := A^{m} B A^{−m}:

W (A, B) = W_{1}(A_{n_{1}}, A_{n_{2}}, …, A_{n_{s_{1}}}) (n_{i} ∈ I ⊂ Z) (2.17)
W (A, B) = W_{2}(B_{m_{1}}, B_{m_{2}}, …, B_{m_{s_{1}}}) (m_{i} ∈ J ⊂ Z) (2.18)

Lemma 2.2 W (A, B) is in Gr^{{d_{1},d_{2}}} ((d_{1}, d_{2}) > (1, 1)) if and only if :

log W_{1}(e^{a^{n_{1}}}, …, e^{a^{ns_{1}}} ) = O(a^{d_{1}})  (2.19)
log W_{2}(e^{b^{m_{1}}}, …, e^{b^{ms_{2}}} ) = O(b^{d_{2}})  (2.20)

The symbol O means that the right-hand sides are of total degree ≥ d_{1} (or d_{2}) in the a_{n_{i}} (or b_{m_{j}}) regarded as free independent variables. The lemma directly follows from the fact that within the algebra \overline{Lie} \{a, b\} (i.e. in the ‘closure’ derived from Lie {a, b} by allowing infinite sums), the subalgebras Lie {a_{n_{1}}, …, a_{n_{s_{1}}}} or Lie {b_{m_{1}}, …, b_{m_{s_{2}}}} finitely generated by a finite number of distinct elements a_{n} or b_{m}:

a_{n} := e^{nb} a e^{−nb} = (exp(n \overline{b})) a (n ∈ Z)  (2.21)
b_{m} := e^{ma} b e^{−ma} = (exp(m \overline{a})) b (m ∈ Z)  (2.22)

are themselves free. So, checking that W (A, B) has alternance (d_{1}, d_{2}) involves only a finite number of steps which, however, steeply increases with s_{1} and s_{2}.

2.3 Low-complexity alternators. Measures of word complexity.

Let us apply the above criterion to words of the form:

W (A, B) = ∏_{j=1}^{r} B^{n_{j}} A B^{n′_{j}} A^{−1} = B^{n_{1}} B_{1}^{n′_{1}} ⋯ B^{n_{r}} B_{1}^{n′_{r}} (n_{j}, n′_{j} ∈ ℤ^{∗}) (2.23)

with indices j regarded as elements of ℤ_{r} := ℤ / r ℤ. For r = 3, we find that W (A, B) has alternance ≥ (1, 3) iff :

∑ n_{j} = ∑ n′_{j} = 0 and n′_{j}/n_{j+2} = λ = const ∈ ℚ (∀j)  (2.24)

and alternance (2, 3) iff λ = 1.

Similarly, for r = 4, the word W (A, B) has alternance ≥ (1, 4) iff :

n_{j} + n_{j+2} = n′_{j} + n′_{j+2} = 0 (∀j)  (2.25)
and n_{j} + n_{j+1}/n′_{j} = τ = const ∈ ℚ (∀j)

and alternance (2, 4) iff τ = −1. In the case r = 3, λ = 1, we may take: (n_{1}, n_{2}, n_{3}) = (n′_{2}, n′_{3}, n′_{1}) = (−1, −1, 2) in which case the (2, 3)-alternator W (A, B) has a fairly simple commutator factorisation:

W (A, B) = {{A, B}, {B {B^{−1}, A}}  (2.26)

but otherwise the simplest words with alternance (d_{1}, d_{2}) tend to be much simpler than their simplest expressions in terms of multicommutators (of the same alternance).

There exist of course many ways of measuring the “complexity” of a word W (A, B) = ∏(A^{m_{i}} B^{n_{i}}), but we shall use only the following four complexity indices:

(i) the length, global or partial:

l(W) = ∑ | m_{i} | + ∑ | n_{i} |
l_{A}(W) = ∑ | m_{i} | ; l_{B}(W) = ∑ | n_{i} |

(ii) the number of alternating monomials A^{m_{i}} or B^{m_{i}}

(iii) the variance var(W) (resp. var_{A}(W) or var_{B}(W)), defined as the number of distinct pairs (m̌_{i}, ň_{i}) (resp. as the number of distinct values assumed separately by m̌_{i} or ň_{i}) whereby m̌_{j} := m_{1}+⋯ m_{j} and ň_{j} := n_{1} + ⋯ n_{j}

(iii) the alternance alt(W) (resp. (alt_{A}(W), alt_{B}(W))) defined as the smallest d (resp. (d_{1}, d_{2})) such that W be in Gr^{{d}} (resp. Gr^{{d_{1},d_{2}}}).

In the usual graphical representation of words, the first three complexity indices are immediate to detect, but not so the alternance, which doesn’t ‘meet the eye’, at least when d_{1} + d_{2} ≥ 3.

2.4 Periodic automorphisms of Gr

Let Aut(Gr) denote the group of all automorphisms of Gr. It is known (Nielsen’s theorem) to be generated by three elementary automorphisms :

(A, B) ⟼ (B, A) ; (A, B) ⟼ (A^{−1}, B) ; (A, B) ⟼ (A B, A) (2.27)

For the sequel, we need to know, up to conjugacy, all the finite subgroups of Aut(Gr) and in particular all the periodic (unipotent) elements of Aut(Gr).

Lemma 2.3 Periodic automorphisms of Gr: They necessarily have order 1,2,3,4, the number of distinct conjugacy classes being respectively 1,4,1,1. Each involution (order 2) is conjugate either to (A, B) ⟼ (B, A) or (A, B) ⟼ (A^{−1}, B) or (A, B) ⟼ (A^{−1}, B^{−1}) or (A, B) ⟼ (B^{−1}, A^{−1}). Each automorphism of order 3 is conjugate to (A, B) ⟼ (B^{−1}, A B^{−1}). Each automorphism of order 4 is conjugate to (A, B) ⟼ (B^{−1} A).

For a proof, see [L.S.], prop.4.6, p25, and the references thereafter.

Now that we have the list of all periodic automorphisms, it is an easy matter to construct all non-cyclic finite subgroups of Aut(Gr). There are four of them (upto conjugacy), namely the two abelian groups of order 4 :

Aut_{1} ≡ {I, I_{1}, I_{2}, I_{3}} (two generators)  (2.28)
Aut_{2} ≡ {I, S_{1}, S_{2}, I_{3}} (two generators)  (2.29)

and the two non-abelian group of order 6 and 8 :

Aut_{3} ≡ {I, S_{1}, J, J^{2}, S_{1} J, J S_{1}} (two generators)  (2.30)
Aut_{4} ≡ {I, I_{1}, I_{2}, I_{3}, S_{1}, S_{2}, K_{1} K_{2}} (two gen., e.g. I_{1} , S_{1}) (2.31)

whose elements P are listed below, in terms of their action P : W ⟼ W^{P} on words :

W^{I}(A, B) = W (A, B) ; W^{I^{1}}(A, B) = W (A^{−1}, B)  (2.32)
W^{I^{2}}(A, B) = W (A, B^{−1}) ; W^{I^{3}}(A, B) = W (A^{−1}, B^{−1}) (2.33)
W^{S^{1}}(A, B) = W (B, A) ; W^{S^{2}}(A, B) = W (B^{−1}, A^{−1}) (2.34)
W^{K^{1}}(A, B) = W (B^{−1}, A) ; W^{K^{2}}(A, B) = W (B, A^{−1})  (2.35)
W J (A, B) = W (B^{−1}, A B^{−1}) ; W^{J^{2}}(A, B) = W (B A^{−1}, A^{−1}) (2.36)
W^{S^{1} J} (A, B) = W (A B, B^{−1}) ; W^{J S^{1}}(A, B) = W (A^{−1} B, A^{−1}) (2.37)

Note that the composition law follows from the rule (W^{P^{1}})^{P^{2}} = W^{(P_{1} P_{2})} (beware of the proper order). Thus:

I_{1}^{2} = I_{2}^{2} = I_{3}^{2} = S_{1}^{2} = S_{2}^{2} = J^{3} = K_{1}^{4} = K_{2}^{4} = I  (2.38)
K_{1}^{2} = K_{2}^{2} = I_{3} ; I_{σ(1)} I_{σ(2)} I_{σ(3)} = I  (∀ σ ∈ P erm{1, 2, 3}) (2.39)
K_{1} = I_{1} S_{1} = S_{2} I_{1} = S_{1} I_{2} = S_{2} I_{2}  (2.40)
K_{2} = S_{1} I_{1} = I_{1} S_{2} = I_{2} S_{1} = S_{1} I_{1}  (2.41)

3 Identity-tangent twins: twenty typical examples.

3.1 Collapsors and divisors

Our aim in this section is to produce examples of identity-tangent twins that cover all the main possible situations and yet remain easy to construct. But this ease of construction comes at a price : it usually entails severe complexity. It also involves highly non-generic relations W (F, G) = 1, with the corresponding words W (A, B) never ranging over subsets of finite codimension in Gr{A, B}. For truly generic examples and finite codimensions, we shall have to wait for §5.

As in most of this paper, we shall have to work simultaneously in the four structures:

\{ \begin{matrix}W (A, B) & ∈ & Gr\{A, B\} & ⟶ & G̃_{0} & ∋ F \\ & & ↓ & & ↓↑ \\ w(a, b) & ∈ & \overline{Lie}\{a, b\} & ⟶ & L̃_{0} & ∋ F_{∗} \end{matrix}  (3.1)

G̃_{0} is the group of formal, identity-tangent diffeos f : x ⟼ x {1 + ∑ a_{n} x^{n}}, usually denoted by the corresponding substitution operators F. L̃_{0} is the Lie algebra of G̃_{0}, with its natural basis elements:

l_{n} := x^{n+1} ∂ := x^{n+1} ∂/∂_{x} (n = 1, 2, …)  (3.2)

The map F ⟼ F_{∗} := log F from G̃_{0} into L̃_{0} is one-to-one, but the injective map W (A, B) ⟼ w(a, b) = log W (e^{a}, e^{b}) from Gr{A, B} into \overline{Lie}\{a, b\} is of course far from surjective. Among the maps from \overline{Lie}\{a, b\} into L̃_{0}, deserving of special attention are the graded morphisms (ie which respect the natural graduation of both algebras) and are necessarily of the form :

(a, b) ⟼ (l_{p}, l_{q}) (p, q ∈ ℕ^{∗})  (3.3)

Perhaps the quickest way to produce twins in G̃_{0} is to consider multicommutators W (A, B) (with alternance (d_{1}, d_{2}) > (1, 1)) whose corner component w_{0}(a, b) (defined as the homogeneous component of w(a, b) that lies in Lie^{[d_{1},d_{2}]}) has the following three properties:

(i) w_{0}(a, b) ≠ 0.

(ii) It is a collapsor, i.e. it belongs to the kernel of some (or all) gradued morphisms of \overline{Lie} into L̃_{0}:

w_{0}(l_{p}, l_{q}) = 0 for some (resp. all ) p, q in ℕ^{∗} × ℕ^{∗}  (3.4)

(iii) Its divisor D is ≠ 0 for all t in N^{∗}. The divisor D(t) ≡ D(p, q; t) is a polynomial in t characterised (for all p, q such that w_{0}(l_{p}, l_{q}) = 0) by the identity:

w_{0}(l_{p} + ϵ_{1} l_{p+t}, l_{q} + ϵ_{2} l_{q+t}) ≡
(ϵ_{1} (q − t) − ϵ_{2} (p − t)) D(t) l_{p d_{1}+q d_{2}+t} + o(ϵ_{1}, ϵ_{2})  (3.5)

which makes good sense, since the right-hand of (3.5) is a priori of the form:

(ϵ_{1} D_{1}(t) − ϵ_{2} D_{2}(t)) l_{p d_{1}+q d_{2}+t} + o(ϵ_{1}, ϵ_{2})

but necessarily vanishes when we take:

ϵ_{1} l_{p+t} = ϵ [l_{t}, l_{p}] = ϵ (p − t) l_{p+t}
ϵ_{2} l_{q+t} = ϵ [l_{t}, l_{q}] = ϵ (q − t) l_{q+t}

Divisors owe their name to the fact that, when solving the equation W (F, G) = 1 through coefficient identification, one has to divide by D(t) at the t-th inductive step. And the ‘collapse’ (3.4) is necessary to get the induction started.

Now, a cheap (if wasteful) way to get words W with those three properties is to take them of the form W = {U, V }, where U and V are themselves multicommutators with non-proportional corner components u_{0}, v_{0} and with distinct (resp. identical) alternance which generally leads to twins with a fixed (resp. free) tangency ratio p/q.

The actual expression of the divisors shall of course involve the structure constants T_{•} of the algebra L_{0}:

[l_{n_{r}} ⋯ [l_{n_{2}}, l_{n_{1}}] ⋯] ≡ T_{n_{1},n_{2},... ,n_{r}} l_{n_{1}+···n_{r}} (l_{n} ≡ x^{n+1} ∂)  (3.6)

which are explicitely given by:

T_{n_{1},n_{2},... ,n_{r}} = (n_{1} − n_{2}) (n_{1} + n_{2} − n_{3}) ⋯ (n_{1} + ⋯ + n_{r−1} − n_{r}) (3.7)

We shall also make frequent use of the identity:

∑_{i}(t − n_{i}) T_{n_{1},... ,n_{i}+t,... ,n_{r}} ≡ (t − (n_{1} + ⋯ + n_{r})) T_{n_{1},... ,n_{r}}  (3.8)

which immediately results from applying the derivation ad(l_{t}) to both sides of (3.6).

3.2 Normal-conormal form of twins. Connectors.

Since knowing a pair of twins (F, G) is the same as knowing their infinitesimal generators (F_{∗}, G_{∗}), we shall often work with the latter, for the greater flexibility they afford. And since these pairs are defined upto a common conjugacy, we shall priviledge “normal-conormal forms”, which normalise one of the twins to F_{∗}^{nor} (resp. G_{∗}^{nor}) while the other assumes a rigidly fixed conormal form G_{∗}^{conor} (resp. F_{∗}^{conor}). Actually, for complete rigidity we have to demand that F_{∗}^{conor} and G_{∗}^{conor} should contain no term of the form l_{p+q} if F_{∗} starts with l_{p} and G_{∗} with l_{q}.

In the most typical examples, these namely with exactly one parameter (other than the ratio p/q, which is discrete), we shall find that :

F_{∗}^{nor} = f_{∗}^{nor}(x) ∂ = (1 − α α_{∗} x^{p})^{−1} (\frac{α}{p} x^{1+p} ∂)  (3.9)
G_{∗}^{conor} = g_{∗}^{conor}(x) ∂ = (1 + β β_{∗} x^{q} +
∑_{m,n≥0 , m+n≥2} α_{(m,n)} α^{m} β^{n} x^{m p+n q}) (\frac{β}{q} x^{1+q} ∂)  (3.10)

F_{∗}^{conor} = f_{∗}^{conor}(x) ∂ = (1 + α α_{∗} x^{p} +
∑_{m,n≥0 , m+n≥2} β_{(m,n)} α^{m} β^{n} x^{m p+n q}) (\frac{α}{p} x^{1+p} ∂)  (3.11)

G_{∗}^{nor} = g_{∗}^{nor}(x) ∂ = (1 − β β_{∗} x^{q})^{−1} (\frac{β}{q} x^{1+q} ∂)  (3.12)

with a countable infinity of invariants:

(α, β) , (α_{∗}, β_{∗}) , (p, q) , {α_{(m,n)} ; m, n ∈ ℕ^{∗}} or {β_{(m,n)} ; m, n ∈ ℕ^{∗}} (3.13)

Another object of central importance, on account both of its invariance and (anti)symmetry in (F, G), is the connector H^{nor} = exp H_{∗}^{nor} which conjugates the normal to the conormal forms:

F^{conor} = H^{nor} F^{nor} (H^{nor})^{−1}  (3.14)
G^{conor} = (H^{nor})^{−1} G^{nor} H^{nor}  (3.15)

but which may also be constructed directly from any solution (F, G) by setting :

F = H_{1} F^{nor} H_{1}^{−1} ; G = H_{2} G^{nor} H_{2}^{−1}  (3.16)
H^{nor} = H_{2}^{−1} H_{1}  (3.17)

The connector H^{nor} is uniquely defined, provided we normalise it by imposing the absence in H_{∗}^{nor} of terms l_{p} and l_{q}. In the most typical (one-continuous parameter) examples, we shall see that the connector assumes the form :

H_{∗}^{nor} = (1 + \begin{aligned}& ∑ \\ \{ & \begin{matrix}m, n ≥ 0 \\ m + n ≥ 0 \end{matrix} \end{aligned} γ_{(m,n)} α^{m} β^{n} x^{m p+n q}) ((\frac{1}{p} − \frac{1}{q}) x ∂) (3.18)

In this case, the secondary invariants {α_{(m,n)}} or {β_{(m,n)}} or {γ_{(m,n)}}, together with the primary invariants (α, β) and (α_{∗}, β_{∗}) and (p, q), constitute a complete and free set of joint invariants for the pair (F, G). The three sets of secondary invariants are clearly equivalent. Indeed for m + n ≥ 2:

α_{(m,n)} = (\frac{1}{p} − \frac{1}{q}) (m p + n q − p) γ_{(m,n)} + earlier terms (3.19)
β_{(m,n)} = (\frac{1}{q} − \frac{1}{p}) (m p + n q − q) γ_{(m,n)} + earlier terms (3.20)

α_{(m,n)} = −\frac{(m p + n q − p)}{(m p + n q − q)} β_{(m,n)} + earlier terms  (3.21)

with the condition m + n ≥ 2 excluding the terms :

α_{(1,0)} = α_{∗} ; α_{(0,1)} = 0 ; β_{(1,0)} = 0 ; β_{(0,1)} = β_{∗}

and ensuring that the leading factors in (2.19), (2.20), (2.21) be ≠ 0, ∞.

The invariants (α, β) in the first pair of primary invariants have a separate existence only if we restrict ourselves to identity-tangent conjugacies. If we accept conjugacies with a multiplier k′(0) ≠ 1, then there remains only one invariant, namely the ratio:

c = α^{q_{∗}} β^{−p_{∗}} with p_{∗}/q_{∗} = p/q and p_{∗} q_{∗} coprime.  (3.22)

Similarly, the tangency orders (p, q) lose their invariance if we accept ramified conjugacies and there only remains the pair (p_{∗}, q_{∗}) of coprime numbers proportional to (p, q). On the other hand, the third pair (α_{∗}, β_{∗}), as well as all the secondary invariants {α_{(m,n)}} {β_{(m,n)}}, possess strong invariance, since they remain unaffected even by non-identity-tangent or ramified conjugacies.

3.3 Main types of twin-begetting relators W (A, B).

Our first series of examples (1 through 6) illustrates two basic dichotomies :

– the twins (F, G) verifying a given relation W (F, G) = 1 may have a fixed or free tangency ratio p/q

– or again, their continuous parameter c (constructed from the leading coefficients : see (3.22)) may be fixed or free.

The second series of examples (7 through 17) imposes additional symmetries on twins. More precisely, for each of the 10 basic finite groups Aut_{i} of automorphisms of Gr{A, B}, we construct relations W (F, G) ≡ 1 whose set of non-trivial solutions (F, G) is globally invariant under Aut_{i}.

Our third series of examples (18 through 20) deals with more exceptional situations, e.g. with relations W (F, G) = 1 whose general solution (F, G) depends on several continuous parameters with fixed or variable positions (i.e. parameters making their “first appearance” inside coefficients of fixed or variable depth t). They also illustrate related questions such as the glueing or splitting of relators.

3.4 Basic examples.

Example 1 . Fixed ratio p/q, no continuous parameter. If we set W := U^{P} V^{Q} with P, Q ∈ ℤ^{∗} and

U = U(A, B) := \overline{\overline{A}}^{3} B = \{A\{A\{A, B\}\}\}  (3.23)
V = V (A, B) := \overline{\overline{B}}^{2} A = \{B\{B, A\}\}\}  (3.24)

then for each s ∈ ℕ^{∗}, W (F, G) = 1 has a unique (upto conjugacy) twin solution (F, G) with tangency orders (p, q) = (s, 2 s) and fixed invariants c, α_{∗}, β_{∗}.

Proof: Let us look for a solution (F, G) with tangency orders p ≠ q. We may write the infinitesimal generators (F_{∗}, G_{∗}) in the form :

F_{∗} = (1 + ∑_{1≤t} α_{t} x^{t}) (\frac{α}{p} x^{p} ∂) with α ≠ 0  (3.25)

G_{∗} = (1 + ∑_{1≤t} β_{t} x^{t}) (\frac{β}{q} x^{q} ∂) with β ≠ 0  (3.26)

Then if we set:

U_{∗} := log U(F, G) ; V_{∗} := log V (F, G) ; W_{∗} := log W (F, G)  (3.27)

we find by the Campbell-Hausdorff-like identity (1.20):

U_{∗} = (∑_{t≥0} u_{t} x^{t}) (\frac{α^{3} β}{p^{3} q} x^{1+M} ∂) with M = 3 p + q  (3.28)

V_{∗} = (∑_{t≥0} v_{t} x^{t}) (\frac{α β^{2}}{p q^{2}} x^{1+N} ∂) with N = p + 2 q  (3.29)

Moreover, since U and V have as corner components :

u_{0}(a, b) = (\overline{a})^{3} b ; v_{0}(a, b) = (\overline{b})^{2} a  (3.30)

the series in U_{∗} and V_{∗} have initial coefficients :

u_{0} = T_{q,p,p,p}  (3.31)
v_{0} = T_{p,q,q}  (3.32)

and current coefficients of the form :

u_{t} = u_{t}^{∗} α_{t} + u_{t}^{∗∗} β_{t} + earlier terms  (3.33)
v_{t} = v_{t}^{∗} α_{t} + v_{t}^{∗∗} β_{t} + earlier terms  (3.34)

with

u_{t}^{∗} = T_{q,p+t,p,p} + T_{q,p,p+t,p} + T_{q,p,p,p+t}  (3.35)
u_{t}^{∗∗} = T_{q+t,p,p,p}  (3.36)
v_{t}^{∗} = T_{p+t,q,q}  (3.37)
v_{t}^{∗∗} = T_{p,q+t,q} + T_{p,q,q+t}  (3.38)

In view of (2.28); (2.29) the identity W (F, G) = 1 clearly implies M = N, which in turn implies (p, q) = (s, 2 s). It also implies the vanishing of the first coefficient in W_{∗}:

P u_{0} \frac{α^{3} β}{p^{3} q} + Q v_{0} \frac{α β^{2}}{p q^{2}} = 0  (3.39)

which rigidly fixes the invariant c:

c := α^{2} β^{−1} = −\frac{p^{2}}{q} \frac{Q}{P} \frac{u_{0}}{v_{0}} = +\frac{p Q}{(p + q) P} = +\frac{Q}{3 P}  (3.40)

As for the current coefficients (α_{t}, β_{t}), their first occurence in W_{∗} takes place at depth t. More precisely :

W_{∗} = (∑_{1≤t} w_{t} x^{t}) (\frac{α^{4} β^{3}}{p^{4} q^{3}} x^{1+5 s})  (3.41)

with

w_{t} = w_{t}^{∗} α_{t} + w_{t}^{∗∗} β_{t} + earlier terms  (3.42)
w_{t}^{∗} = P \frac{α^{3} β}{p^{3} q} u_{t}^{∗} + Q \frac{α β^{2}}{p q^{2}} v_{t}^{∗}  (3.43)

w_{t}^{∗∗} = P \frac{α^{3} β}{p^{3} q} u_{t}^{∗∗} + Q \frac{α β^{2}}{p q^{2}} v_{t}^{∗∗}  (3.44)

But due to (3.29) we may factor out the term:

R = −P u_{0} \frac{α^{3} β}{p^{3} q} = −Q v_{0} \frac{α β^{2}}{p q^{2}}  (3.45)

Eventually, after expliciting u_{0},v_{0} etc... and recalling that (p, q) = (s, 2 s), we find

w_{t}^{∗} = R (−\frac{u_{t}^{∗}}{u_{0}} + \frac{v_{t}^{∗}}{v_{0}} ) = R \frac{1}{6} (6 + t + t^{2}) (t − q)  (3.46)
w_{t}^{∗∗} = R (−\frac{u_{t}^{∗∗}}{u_{0}} + \frac{v_{t}^{∗∗}}{v_{0}} ) = \frac{1}{6} R (6 + t + t^{2}) (t − p)  (3.47)

The presence of a common factor (6 + t + t^{2}) alongside the individual factors (t − q) and (t − p) is no accident, but a consequence of the identities :

(t − p) u_{t}^{∗} + (t − q) u_{t}^{∗∗} = t − M  (3.48)
(t − p) v_{t}^{∗} + (t − q) v_{t}^{∗∗} = t − N  (3.49)

which are but special cases of (3.8). Thus in the end we have:

w_{t} ≡ R (1/6) (6 + t + t^{2}) [(t − q) α_{t} − (t − p) β_{t}] + earlier terms (3.50)

Therefore, since solving W (F, G) = 1 means solving W_{∗}(F, G) = 0, i.e. killing all coefficients w_{t}, we have one choice at each inductive step. For t = 0, w_{0} = 0 is ensured by (3.39). For t ∉ {0, p, q}, we may choose α_{t} (or β_{t}) as we wish, after which the other coefficient is fixed. For the exceptional values p and q, however, the choice is restricted to one of the coefficients. For t = p, β_{p} can be anything, but α_{p} is rigidly set and equal to α α_{∗} irrespective of the choice for β_{p}. For t = q, α_{q} can be anything, but β_{q} is rigidly set and equal to β β_{∗}. This comes as no surprise, since the ‘iteration residues’ (α_{∗}, β_{∗}) are strongly invariant under conjugations. Their actual values, as given in (3.82) below, result from a general calculation that will be carried out in Example 4.

If we now analyse the ‘earlier terms’ in (3.50) with the help of formula (1.20) applied successively to U(F, G), V (F, G), W (F, G), we find that these ‘earlier terms’ are generated by multiple Lie brackets involving only monomials l_{t_{i}} for earlier indices t_{i} < t, starting with l_{p} and l_{q} (recall that l_{n} := x^{n+1} ∂). We may therefore restrict ourselves to indices t in p ℕ or q ℕ, and set all other coefficients α_{t} or β_{t} equal to 0. In particuliar, if we take advantage of the latitude afforded by the induction to normalise either F_{∗} or G_{∗}, then the conormal twin (see (3.10) (3.11)) will carry only indices of depth t in p ℕ + q ℕ and so too will the connector H^{nor}.

Example 2 . Fixed ratio p/q = p_{0}/q_{0} , one discrete parameter. If we set

W := U^{P} V^{Q} with P, Q ∈ ℤ^{∗} and

U = U(A, B) := (\overline{\overline{A}})^{1+q_{0}} B = \{A\{⋯ A, B\} ⋯ \}\}  (3.51)
V = V (A, B) := (\overline{\overline{B}})^{1+p_{0}} A = \{B\{⋯ B, A\} ⋯ \}\}  (3.52)

then W (F, G) = 1 has a twin solution with tangency orders (p, q) iff p/q = p_{0}/q_{0} (≠ 1). That solution depends on a prime unit root ϵ of order s_{0} (s_{0} being the largest common divisor of p, q).

Proof: Taking F_{∗} and G_{∗} as in (3,25), (3;26), we get this time:

U_{∗} = (∑_{t≥0} u_{t} x^{t}) (\frac{α^{1+p_{0}} β}{p^{1+q_{0}} q} x^{1+M} ∂)  (3.53)

V_{∗} = (∑_{t≥0} v_{t} x^{t}) (\frac{α β^{1+q_{0}}}{p q^{1+p_{0}}} x^{1+N} ∂)  (3.54)

with

M = p (1 + q_{0}) + q ; N = q (1 + p_{0}) + p  (3.55)

The initial coefficients are:

u_{0} = T_{q,p^{[1+q_{0}]}}  (3.56)
v_{0} = T_{p,q^{[1+p_{0}]}}  (3.57)

where of course m^{[n]} denotes the sequence (m, …, m) of length n. The subsequent coefficients u_{t}, v_{t} retain their earlier expression (3.33) (3.34) but with:

u_{t}^{∗} = ∑ T_{q,p^{[q_{1}]},p+t,p^{[q_{2}]}} (for q_{1}, q_{2} ≥ 0 ; q_{1} + q_{2} = q_{0})  (3.58)
u_{t}^{∗∗} = T_{q+t,p^{[1+q_{0}]}}  (3.59)
v_{t}^{∗} = T_{p+t,q^{[1+p_{0}]}}  (3.60)
v_{t}^{∗∗} = ∑ T_{p,q^{[p_{1}]},q+t,q^{[p_{2}]}} (for p_{1}, p_{2} ≥ 0 ; p_{1} + p_{2} = p_{0})  (3.61)

But the earlier identities (3.48) (3.49) remain in force (being a simple consequence of (3.8)) and lead to a drastic simplification of u_{t}^{∗} and v_{t}^{∗∗}:

u_{t}^{∗} = \frac{t − M}{t − p} − \frac{t − q}{t − p} T_{q+t,p^{[1+q_{0}]}}  (3.62)
v_{t}^{∗∗} = \frac{t − N}{t − q} − \frac{t − p}{t − q} T_{p+t,q^{[1+p_{0}]}}  (3.63)

Here too, W_{∗} may vanish only if M = N, which in view of (3.55) imposes p/q = p_{0}/q_{0}. So we may write :

(p, q) = (s p_{∗}, s q_{∗}) ; (p_{0}, q_{0}) = (s_{0} p_{∗}, s_{0} q_{∗})  (3.64)

with s, s_{0} in N^{∗} and p_{∗}, q_{∗} coprime. The argument then proceeds as in Example 1, except that (3.39) now becomes :

P u_{0} (\frac{α β}{p q}) (\frac{α}{p})^{s_{0} p_{∗}} + Q v_{0} (\frac{α β}{p q}) (\frac{β}{q})^{s_{0} q_{∗}} = 0  (3.65)

so that the invariant c := α^{q_{∗}} β^{−p_{∗}} is now constrained only by :

c^{s_{0}} = −\frac{Q v_{0}}{P u_{0}} \frac{p^{s_{0} p_{∗}}}{q^{s_{0} q_{∗}}}  (3.66)

and so depends on a unit root ϵ of order s_{0}. That aside, the key identity (3.50) has its exact counterpart :

w_{t} = R . D(t) [(t − q) α_{t} − (t − p) β_{t}] + earlier terms  (3.67)

with a trivial factor :

R = −P u_{0} (\frac{α β}{p q}) (\frac{α}{p})^{s_{0} p_{∗}} = Q v_{0} (\frac{α β}{p q}) (\frac{β}{q})^{s_{0} q_{∗}}  (3.68)

and a non-trivial divisor D(t) = T_{q,p^{[1+q_{0}]}} T_{p,q^{[1+p_{0}]}} Δ(t), with

Δ(t) = Δ_{p,q;p_{0},q_{0}}(t) = (t − p)^{−1} (t − q)^{−1} (−(t − S) +
(t − q) \frac{T_{q+t,p^{[1+q_{0}]}}}{T_{q,p^{[1+q_{0}]}}} + (t − p) \frac{T_{p+t,q^{[1+p_{0}]}}}{T_{p,q^{[1+p_{0}]}}} ) (3.69)

and

S = M = N = s s_{0} p_{∗} q_{∗} + s p_{∗} + s q_{∗}  (3.70)

Here again, despite the denominator (t − p) (t − q), D(t) is necessarily a polynomial in t. Thus, if we take s = s_{0} = 1 and therefore (p, q) = (p_{0}, q_{0}) = (p_{∗}, q_{∗}), we find :

D(t) = Const (t^{3} + 15 t^{2} + 26 t + 120) for (p_{∗}, q_{∗}) = (1, 3)
D(t) = Const (2 t^{3} + 15 t^{2} + 55 t + 210) for (p_{∗}, q_{∗}) = (2, 5)
D(t) = Const (t^{4} + 26 t^{3} + 271 t^{2} + 606 t + 2520) for (p_{∗}, q_{∗}) = (1, 4)

etc... The remarkable thing is that, although (t − p) (t − q) D(t) is a real polynomial of mixed signs, D(t) itself seems always to carry positive coefficients only, which automatically ensures D(t) ≠ 0) for t ∈ ℕ^{∗} and thus guarantees the solvability of the relation W (F, G) = 1. It should be pointed out, however, that this positivity result has not been proved for all values p/q = p_{0}/q_{0}, but only checked, on a case by case basis, for all pairs (p, q), (p_{0}, q_{0}) with p + q ≤ 30, p_{0} + q_{0} ≤ 30. However, reverting to the expression (3.7) of T_{•}, we may rewrite Δ(t) as:

Δ(t) =  Δ_{p,q;p_{0},q_{0}}(t) = \frac{\frac{1}{2} (p_{0} q+p q_{0})+(p+q)−t}{(t−p) (t−q)}
+\frac{1}{t−q} \frac{Γ( \frac{p+t}{q} +p_{0}) Γ( \frac{p}{q} −1)}{Γ( \frac{p}{q} +p_{0}) Γ( \frac{p+t}{q} −1)} + \frac{1}{t−p} \frac{Γ( \frac{q+t}{p} +q_{0}) Γ( \frac{q}{p} −1)}{Γ( \frac{q}{p} +q_{0}) Γ( \frac{q+t}{p} −1)}  (3.71)

Then,using the asymptotic properties of the gamma function, it is a straightforward exercise to check that, for any fixed pair (p, q) and for (p_{0}, q_{0}) = (σ p, σ q) the divisor Δ(t) is > 0 for all t > 0 and all σ large enough. Thus, for any given tangency orders (p, q), we can point to an explicit relation W (F, G) = 1 that admits twins (F, G) with that tangency order.

Example 3 . Fixed ratio p/q, one continuous parameter. If we set W (A, B) := {U(A, B), V (A, B)} with U,V as in Example 1, then the relation W (F, G) = 1 has twin solutions for the tangency ratios p/q = 1/2. These twins still have the same iteration residues α_{∗} = 15/8 and β_{∗} = −7/12 as in Example 1, but they now depend on a free continuous parameter c := α^{2} β^{−1}

Proof: Starting from any pairs (p, q) and (α, β), we take (F_{∗}, G_{∗}) as in Example 1 and get the same expression for (U_{∗}, V_{∗}). The expression for W_{∗}, however, does change:

W_{∗} = (∑_{0≤t} w_{t} x^{t}) (\frac{α β}{p q})(\frac{α}{p})^{2} (\frac{β}{q}) (x^{1+M+N} ∂)  (3.72)

with an initial coefficient w_{0} that vanishes iff M = N. Thus W_{∗} = 0 implies p/q = 1/2 as before but leaves (α, β) and thus c := α^{2} β^{−1} completely free. As for the current coefficient w_{t}, it is now of the form :

w_{t} = D(t) ((t − q) α_{t} − (t − p) β_{t}) + earlier terms  (3.73)

with the same divisor D(t) as in Example 1, upto the trivial factor u_{0} v_{0}/R :

D(t) = u_{0} v_{0} t \frac{1}{6} (6 + t + t^{2}) with u_{0} = T_{q,p,p} , v_{0} = T_{p,q,q}  (3.74)

So the only change is the appearance of the continuous parameter c (invariant under general conjugacies) or the pair (α, β) (invariant under identity-tangent conjugacies) instead of the rigidity in Example 1. As for the values (α_{∗}, β_{∗}), they are special cases of (3.76),(3.77) below.

Example 4 . Fixed ratio p/q = p_{0}/q_{0} , one continuous parameter. If we set W (A, B) := {U(A, B), V (A, B)} with U,V as in Example 2

(i.e. U := (\overline{\overline{A}})^{1+q_{0}} B and V := (\overline{\overline{B}})^{1+p_{0}} A) then the relation W (F, G) = 1 has twin solutions for the tangency ratios p/q = p_{0}/q_{0}. These twins still possess the same iteration residues (α_{∗}, β_{∗}) as in Example 2 (see (3.81) below), but they now depend on a free parameter c := α^{q_{∗}} β^{−p_{∗}}

Proof: The transition is the same as from Example 1 to Example 3. We find this time :

W_{∗} = (∑_{0≤t} w_{t} x^{t}) (\frac{α β}{p q})(\frac{α}{p})^{q_{0}} (\frac{β}{q})^{p_{0}} (x^{1+M+N} ∂)  (3.75)

with M, N as in (3.53). The initial coefficient w_{0} vanishes iff M = N, which implies p/q = p_{0}/q_{0} = p_{∗}/q_{∗} (irreducible) as in Example 2 but leaves (α, β) and thus c := α^{q_{∗}} β^{−p_{∗}} free. The current coefficient w_{t} is still of the form :

w_{t} = D(t) ((t − q) α_{t} − (t − p) β_{t}) + earlier terms  (3.76)

with the same divisor D(t) as in Example 2, upto the trivial factor u_{0} v_{0}/R:

D(t) = u_{0} v_{0} t Δ(t) (with Δ(•) as in (3.69))  (3.77)

That leaves only one point to settle, namely the invariants α_{∗}, β_{∗}. These are known as soon as the ‘earlier terms’ in (3.74) are known for t = p and t = q. To calculate these, we must express u(a, b) := log U(ea, eb) and v(a, b) := log V (ea, eb) and retain not just the terms of lowest degree (i.e. the ‘corner components’) but also the terms of immediately higher degree, by using formula (1.20) (with a_{j} either a or b) and then substituting (l_{p}, l_{q}) for (a, b). Eventually, we find expressions that depend only on (p_{0}, q_{0}) but not (p, q) :

α_{∗} = \frac{1}{2} \frac{(1 + q_{0})^{2} − \frac{T_{p_{0},q_{0},p_{0},q{[p_{0}] \atop 0}}}{p_{0} T_{p_{0},q{[1+p_{0}] \atop 0}}}}{(q_{0} − p_{0}) Δ(p_{0})}  (3.78)

β_{∗} = \frac{1}{2} \frac{(1 + p_{0})^{2} − \frac{T_{q_{0},p_{0},q_{0},p{[q_{0}] \atop 0}}}{q_{0} T_{q_{0},p{[1+q_{0}] \atop 0}}}}{(p_{0} − q_{0}) Δ(q_{0})}  (3.79)

with Δ(•) = Δ_{p,q;p_{0},q_{0}}(•) as in (3.67) and (3.69). Remarkably, these formulas don’t apply directly when t = p_{0} or q_{0} for then the denominators vanish. We must either calculate Δ(t) first as a polynom in t or else resort to the formulas :

(q_{0} − p_{0}) Δ(p_{0}) = \frac{∑ T_{q_{0},p_{0}^{[q_{1}]},2 p_{0},p_{0}^{[q_{2}]}}}{T_{q_{0},p_{0}^{[1+q_{0}]}}} − \frac{T_{2 p_{0},q_{0}^{[1+p_{0}]}}}{T_{p_{0},q_{0}^{[1+p_{0}]}}}  (3.80)

(p_{0} − q_{0}) Δ(q_{0}) = \frac{∑ T_{p_{0},q_{0}^{[p_{1}]},2 q_{0},q_{0}^{[p_{2}]}}}{T_{p_{0},q_{0}^{[1+p_{0}]}}} − \frac{T_{2 q_{0},p_{0}^{[1+q_{0}]}}}{T_{q_{0},p_{0}^{[1+q_{0}]}}}  (3.81)

with a summation extending to all q_{i} ≥ 0 such that q_{1} + q_{2} = q_{0} (resp. to all p_{i} ≥ 0 such that p_{1} + p_{2} = p_{0}. Thus we get:

(α_{∗}, β_{∗}) = (30/29, −7/12)  for (p_{0}, q_{0}) = (1, 2) (3.82)
(α_{∗}, β_{∗}) = (50/27, −8/45)  for (p_{0}, q_{0}) = (1, 3) (3.83)
(α_{∗}, β_{∗}) = (413/132, −625/564) for (p_{0}, q_{0}) = (2, 3) (3.84)
(α_{∗}, β_{∗}) = (1785/856, −69/1120) for (p_{0}, q_{0}) = (1, 4) (3.85)

Example 5 . Free ratio p/q one continuous parameter. If we set W (A, B) := {U(A, B), V (A, B)} with

U = U(A, B) := \overline{\overline{A}} \overline{\overline{B}} \overline{\overline{B}} \overline{\overline{B}} A = \{A\{B\{B\{B, A\}\}\}\}  (3.86)
V = V (A, B) := \overline{\overline{B}} \overline{\overline{A}} \overline{\overline{B}} \overline{\overline{A}} B = \{B\{A\{B\{A, B\}\}\}\}  (3.87)

then the relation W (F, G) = 1 has a twin solution for each p ≠ q and each (α, β). That solution is unique (upto identity-tangent conjugacies), with iteration residues:

α_{∗} = \frac{q (4 p^{2} + 5 p q + 3 q^{2})}{p^{2} (p + 7 q)} ; β_{∗} = \frac{3 p^{2} (p + q)}{q^{2} (3 p + q)}  (3.88)

Proof: In the free structures Gr{A, B} and Lie{a, b}, the elements U,V ,W have alternance (2,3),(2,3),(4,6) and their Lie images u,v,w have corner components :

u_{0} = \overline{a} \overline{b} \overline{b} \overline{b} a ; v_{0} = \overline{b} \overline{a} \overline{b} \overline{a} b ; w_{0} = [u_{0}, v_{0}]  (3.89)

Going over to the bound structures G̃_{0} and L̃_{0} and defining U, V, W and U_{∗}, V_{∗},W_{∗} in the usual way, we find :

U_{∗} = (∑_{t≥0} u_{t} x^{t}) (\frac{α}{p})^{2} (\frac{β}{q})^{3} (x^{1+M} ∂)  (3.90)

V_{∗} = (∑_{t≥0} v_{t} x^{t}) (\frac{α}{p})^{2} (\frac{β}{q})^{3} (x^{1+M} ∂)  (3.91)

W_{∗} = (∑_{t≥0} w_{t} x^{t}) (\frac{α}{p})^{4} (\frac{β}{q})^{6} (x^{1+M+N} ∂)  (3.92)

but with M = N = 2 p + 3 q irrespective of the choice of (p, q). Furthermore, we have leading coefficients :

u_{0} = T_{p,q,q,q,p} ; v_{0} = T_{q,p,q,p,q} ; w_{0} = 0 (∀p, q)  (3.93)

and the current coefficients of the form :

u_{t} = u_{t}^{∗} α_{t} + u_{t}^{∗∗} β_{t} + earlier terms  (3.94)
v_{t} = v_{t}^{∗} α_{t} + v_{t}^{∗∗} β_{t} + earlier terms  (3.95)
w_{t} = w_{t}^{∗} α_{t} + w_{t}^{∗∗} β_{t} + earlier terms  (3.96)

with (α_{t}, β_{t}) denoting the coefficients of (F_{∗}, G_{∗}) of depth t, as in (3.25) (3.26) and with:

u_{t}^{∗} = T_{p+t,q,q,q,p} + T_{p,q,q,q,p+t}  (3.97)
u_{t}^{∗∗} = T_{p,q+t,q,q,p} + T_{p,q,q+t,q,p} +T_{p,q,q,q+t,p}  (3.98)
v_{t}^{∗} = T_{q,p+t,q,p,q} + T_{q,p,q,p+t,q}  (3.99)
v_{t}^{∗∗} = T_{q+t,p,q,p,q} + T_{q,p,q+t,p,q} +T_{q,p,q,p,q+t}  (3.100)

and therefore :

w_{t}^{∗} = t det ( \begin{matrix}u_{0} & v_{0} \\ u_{t}^{∗} & v_{t}^{∗} \end{matrix} ) ; w_{t}^{∗∗} = t det ( \begin{matrix}u_{0} & v_{0} \\ u_{t}^{∗∗} & v_{t}^{∗∗} \end{matrix} )  (3.101)

But applying (3.8) to u_{0} and v_{0} we get :

(t − 2 p − 3 q) u_{0} = (t − p) u_{t}^{∗} + (t − q) u_{t}^{∗∗}  (3.102)
(t − 2 p − 3 q) v_{0} = (t − p) v_{t}^{∗} + (t − q) v_{t}^{∗∗}  (3.103)

As a consequence, (t − p) w_{t}^{∗} + (t − q) w_{t}^{∗∗} = 0 and we may write :

w_{t} = D(t) ((t − q) α_{t} − (t − p) β_{t}) + earlier terms  (3.104)

with a divisor D(t) given by :

D(t) := \frac{t}{t − q} det ( \begin{matrix}u_{0} & v_{0} \\ u_{t}^{∗} & v_{t}^{∗} \end{matrix} ) = −\frac{t}{t − p} det ( \begin{matrix}u_{0} & v_{0} \\ u_{t}^{∗∗} & v_{t}^{∗∗} \end{matrix} )  (3.105)

A simple calculation then yields :

D(t) = p q (p − q)^{2} t^{2} (t^{2} + q t + 6 p q)  (3.106)

so that the divisor D(t) is always ≠ 0 for positive integers (p, q, t) (p ≠ q). The upshot is that, given initial data (p, q) and (α, β) (p ≠ q ; α β ≠ 0), the inductive resolution of {w_{t} = 0} is possible, leading to ‘intrinsical series’ for F_{∗}^{conor}, G_{∗}^{conor}, H_{∗}^{nor} that have exactly the form (3.10), (3.11), (3.18) and carry only coefficients of depth t in p ℕ + q ℕ.

To find the exact values of the iteration residues (α_{∗}, β_{∗}), we must first apply (1.20) to calculate the terms of degree (2 + 1, 3) and (2, 3 + 1) in u(a, b) := log U(ea, eb) and v(a, b) := log V (ea, eb); then plug this into w(a, b) := log W (ea, eb); and lastly replace (a, b) by (l_{p}, l_{q}). We find :

α_{∗} = −\frac{1}{2} \frac{1}{p − q} \frac{(u_{0} v′_{0} − v_{0} u′_{0})}{D(p)} = −\frac{1}{2 p} \begin{matrix}det & ( \begin{matrix}u_{0} & v_{0} \\ u′_{0} & v′_{0} \end{matrix} ) \\ det & ( \begin{matrix}u_{0} & v_{0} \\ u_{p}^{∗} & v_{p}^{∗} \end{matrix} ) \end{matrix}  (3.107)

β_{∗} = −\frac{1}{2} \frac{1}{p − q} \frac{(u_{0} v′′_{0} − v_{0} u′′_{0})}{D(q)} = −\frac{1}{2 q} \begin{matrix}det & ( \begin{matrix}u_{0} & v_{0} \\ u′′_{0} & v′′_{0} \end{matrix} ) \\ det & ( \begin{matrix}u_{0} & v_{0} \\ u_{q}^{∗∗} & v_{q}^{∗∗} \end{matrix} ) \end{matrix}  (3.108)

with u_{0}, v_{0}, D(t) as above and :

u′_{0} = T_{p,q,p,q,q,p} + T_{p,q,q,q,p,p}  (3.109)
u′′_{0} = 3 T_{p,q,q,q,q,p}  (3.110)
v′_{0} = T_{q,p,p,q,p,q} + T_{q,p,q,p,p,q}  (3.111)
v′′_{0} = 2 T_{q,p,q,q,p,q} + T_{q,p,q,p,q,q}  (3.112)

Which eventually leads to the values α_{∗}, β_{∗} mentioned in (3.85).

Remark1 : We observe that α_{∗}, β_{∗}, as indeed all the secondary invariants α_{(m,n)}, β_{(m,n)}, γ_{(m,n)} carried by the “intrinsic series” (3.10),(3.11), (3.18), are homogeneous functions of degree 0 in (p, q).

Remark2 : For the time being, we have set aside the case of twins with identical tangency order p = q. But we may note that when p/q ⟶ 1, then α^{−1} F_{∗} and β^{−1} G_{∗} in Example 5 tend to common, non-trivial series. In particuliar, α_{∗} and β_{∗} go to a common limit, which interestingly is ≠ 0. As (3.85) shows, that limit is 3/2.

Remark3 : What makes Example 5 tick is clearly not the precise shape of U(A, B) and V (A, B), but the following three circumstances:

(i) the words U(A, B) and V (A, B) have the same alternance (d_{1}, d_{2})

(ii) the corner components u_{0}(a, b) and v_{0}(a, b) are independent in Lie^{[d_{1},d_{2}]}

(iii) the divisor D(t) ≡ D(p, q; t) is ≠ 0 for p ≠ q and t > 0.

Example 5 happens to be the simplest possible instance of this type since it corresponds to (d_{1}, d_{2}) = (2, 3), which is the lowest alternance for which Lie^{[d_{1},d_{2}]} has dimension > 1. But in this case, the pair (d_{1}, d_{2}) consists of an even and an odd number, and for the sequel it will be useful to have examples corresponding to the two other combinations: odd+odd, even+even.

Example 6 . Free ratio p/q, one continuous parameter. If we set W := {U, V } with distinct words U, V of equal alternance (d_{1}, d_{2}), the picture is usually the same as in Example 5, for the condition (ii) and (iii) are generically fulfilled. In particular, for (d_{1}, d_{2}) = (3, 3) = (odd, odd), we may take:

U = U(A, B) := \overline{\overline{A}}^{2} \overline{\overline{B}}^{3} A ; V = V (A, B) := \overline{\overline{B}}^{2} \overline{\overline{A}}^{3} B (3.113)

since in this case:

D(t) = 3 p q (p − q)^{2} t^{2} × …  (3.114)
(2 t^{3} + 5 (p + q) t^{2} + (3 p^{2} + 14 p q + 3 q^{2}) t + 24 p q (p + q))

with positive coefficients only. We also find:

α_{∗} = \frac{q (p + 3 q) (18 p^{2} + 23 p q + 6 q^{2})}{p^{2} (10 p^{2} + 43 p q + 27 q^{2})}  (3.115)
β_{∗} = \frac{p (q + 3 p) (18 q^{2} + 23 q p + 6 p^{2})}{q^{2} (10 q^{2} + 43 q p + 27 p^{2})}  (3.116)

Similarly, for (d_{1}, d_{2}) = (2, 4) = (even, even) we may take :

U = U(A, B) := \overline{\overline{A}} \overline{\overline{B}}^{4} A ; V = V (A, B) := \overline{\overline{B}}^{3} \overline{\overline{A}}^{2} B  (3.117)

since in this case :

D(t) = 4 p q (p − q)^{2} (p + q) (6 p q + q t + t^{2}) (2 q + 2 p + t)  (3.118)

again with positive coefficients only. Here again we find: α_{∗} ≠ 0 , β_{∗} ≠ 0.

We skip the proofs, since they follow axactly the same lines as in Example 5. Of course, the above Remarks 1 and 2 still apply.

3.5 Examples with built-in symmetries

To a given twin-generating relation W (F, G) = 1, one may not add any independent relation W_{1}(F, G) = 1 without forcing F and G to commute (which by our definition twins are forbidden to do). One may well, however, impose additional symmetries. More precisely, given any one of the ten (upto conjugacy) finite subgroups ^{3} of Aut Gr{A, B}, one may look for relations R(A, B) = 1 that generate twins while being invariant under the action of the subgroup Aut_{j} in question :

R^{T}(A, B) = Q_{T}(A, B) (R(A, B))^{ϵ_{T}} (Q_{T}(A, B))^{−1}  (3.119)
( ∀T ∈ Aut_{j} ; ϵ_{T} = ±1 )

(Here the exponent ϵ_{T} has to be an integer and a unit root. So ϵ_{T} = ±1).

In each of these ten cases, we shall restrict ourselves to relations R(F, G) ≡ 1 whose general twin solution (F, G) depends only on a free ratio p/q ( ≠ 1 unless stated otherwise) and a free continuous parameter c = α^{q_{∗}} β^{−p_{∗}} (like in Examples 5 or 6). And as always in this section, we shall strive to pick those examples which are easiest to construct, rather than those with the lowest word complexity.

We begin with the six cyclic groups of automorphisms.

Example 7 . Invariance under S_{1} : (A, B) ↦ (B, A).

If we set U := \overline{\overline{A}}^{2} \overline{\overline{B}}^{3} A and so U^{S_{1}} = \overline{\overline{B}}^{2} \overline{\overline{A}}^{3} B, then R := \{U, U^{S^{1}}\} is a twin-generating relation invariant under S_{1} whose general twin solution depends only on p/q and c = α^{q_{∗}} β^{−p_{∗}}.

Proof: This is in fact nothing but Example 6 for (d_{1}, d_{2}) = (3, 3) since then we had V ≡ U^{S^{1}}. So all there is to check is the S_{1} -invariance of the relation R ≡ 1, which simply results from R^{S^{1}} ≡ R^{−1}.

Example 8 . Invariance under S_{2} : (A, B) ↦ (B^{−1}, A^{−1}).

If we take once again U := \overline{\overline{A}}^{2} \overline{\overline{B}}^{3} A and so U^{S^{2}} = (\overline{\overline{B^{−1}}})^{2} (\overline{\overline{A^{−1}}})^{3} B^{−1}, then R := {U, U^{S^{2}}} is a twin-generating relation invariant under S_{2} and with a general solution depending on p/q and c.

3Their list was given at the end of §2, along with the notation for their elements.

We recall that among these 10 finite subgroups 6 are cyclic; 2 abelian non-cyclic; and 2 non-abelian.

Proof: This new relation R not only differs from that of Example 7 but is not even equivalent to it. However, it has the same corner component r_{0}(a, b) and so the same non- vanishing divisor D(t) ( see (3.111)). As for the invariance, it results once again from R^{S^{2}} ≡ R^{−1}.

Example 9 . Invariance under I_{0} : (A, B) ↦ (A^{−1}, B^{−1}). Let W = {U, V } be a twin-generating relation as in Example 5 or 6. Since U, V necessarily share the same (global) alternance d = d_{1}+d_{2}, W is always of even (global) alternance 2 d and W_{1} = {W , A} of odd (global) alternance 2 d + 1. If we then we set:

R := W W^{I^{0}}  (3.120)
R_{1} := W_{1} (W I_{0}_{1})^{−1}  (3.121)

we get two (non-equivalent) twin-generating and I_{0}-invariant relations R and R_{1}, whose general solution depends on p/q and c = α^{q_{∗}} β^{−p_{∗}}, but whose iteration residues (α_{∗}, β_{∗}) automatically vanish.

Proof: The I_{0}-invariance trivially results from :

R^{I^{0}} := W^{−1} R W (even alternance)  (3.122)
RI_{0}_{1} := R_{1}^{−1} (odd alternance)  (3.123)

The corner component and the divisor of R (resp. R_{1}) are twice those of W (resp. W_{1}). The divisor D(p, q; t) of W, in turn, being the same as in Example 5 or 6, does not vanish for p ≠ q and p, q, t ∈ ℕ^{∗}. Neither does the divisor D_{1}(p, q; t) of W_{1} since:

w_{10}(a, b) = [w_{0}(a, b), a]  (3.124)

Hence

D_{1}(p, q; t) = ((d_{1} − 1) p + d_{2} q + t) D(p, q; t)  (3.125)

The simpler invariance relation (3.120) (bought at the cost of a more complex W_{1}) implies that the Lie image r_{1}(a, b) of R(A,B) carries only components of odd (global) degree in (a,b). The component of lowest degree is the corner component w_{10}(a, b), with degree exactly 2 d + 1. But the next two components (of degree 2 d + 2) vanish, and so do the invariants α_{∗}, β_{∗} which stem from these components. The same conclusion also holds for the solutions of the relation R ≡ 1, but the proof is slightly less direct. First, we note that (3.119) implies :

r(−a, −b) = e^{−\overline{w(a,b)}} r(a, b)  (3.126)

So, if we set :

r_{2}(a, b) = r(a, b) + r(−a, −b)  (3.127)

we have :

r_{2}(a, b) = (1 + e^{−\overline{w(a,b)}}) r(a, b)  (3.128)

Since 1 + exp(−\overline{w}), as an operator on \overline{Lie}(a, b) is clearly invertible, the identity r(a, b) ≡ 0 is equivalent to r_{2}(a, b) = 0 (though r_{2}(a, b) is not the Lie image of any word R_{2}(A, B)). Now, in view of its definition (3.125), r_{2}(a, b) carries only components of even (global) degree in (a, b). So here again the two components of r_{2}(a, b) immediately superior (in degree) to the corner component necessarily vanish, so that α_{∗} = β_{∗} = 0.

Example 10 . Invariance under I_{1} : (A, B) ↦ (A^{−1}, B) Let W := {U, V } be a twin-generating relation of alternance (d_{1}, d_{2}) in (A,B), with d_{1} even (as in Example 5) or odd (as in the same Example, but with A and B exchanged). Then if we set :

R := W W^{I^{1}} (for d_{1} even)  (3.129)
R_{1} := W (W^{I^{1}})^{−1} (for d_{1} odd)  (3.130)

we get I_{1}- invariant twin-generating relations. As usual, their general solution depends on p/q and c, but here only α_{∗} is guaranteed to vanish.

Proof: Same as for Example 9.

Example 11 . Invariance under J : (A, B) ↦ (B^{−1}, A B^{−1})

If we set successively :

C := {A, B}  (3.131)
U := \{\overline{\overline{C}}^{2} A, \overline{\overline{C}}^{2} B\}  (3.132)
V := \overline{\overline{C}}^{2} \{\overline{\overline{C}} A, \overline{\overline{C}} B\}  (3.133)
W := {U, V }  (3.134)
R := W W J W^{J^{2}}  (3.135)

then the J-invariant relation R(F, G) ≡ 1 has a general twin-solution that depends on p/q and c.

Proof: The Campbell-Hausdorff formula shows that the action of J on the images of (A, B) is :

(a, b) ↦ (−b, a − b) mod [a, b]  (3.136)

So here the restriction to the terms of lowest degree is not a diagonal mapping. Therefore, if we want to construct an example which, like the previous ones, leads to twins with a completely free ratio p/q, special precautions have to be taken to produce words U, V whose Lie images u, v have corner components u_{0}, v_{0} that are :

(i) non-proportional

(ii) separately invariant under (3.136)

(iii) and which give rise to a non-vanishing divisor.

If furthermore we want U and V to be expressible each as one single multicommutator, then the “simplest” possible choice happens to be (3.132),(3.133), yielding:

u_{0} = [[[a, b], [[a, b], a]], [[a, b], [[a, b], b]]]  (3.137)
v_{0} = [[a, b], [[a, b], [[[a, b], a], [[a, b], b]]]]  (3.138)

The non-proportionality of u_{0}, v_{0} is easily checked by specialising (a,b) to (l_{p}, l_{q}). The invariance (mod [a, b]) under (3.136) is immediate. As for the divisor associated with w_{0} = [u_{0}, v_{0}], a tedious but straightforward calculation yields:

D(p, q; t) = 12 p^{2} q^{2} t (p − q)^{2} (t + 6 p + 6 q) (t + p − q) (t − p + q) (3.139)

so that D(p, q; t) ≠ 0 for p ≠ q and t ∈ p ℕ+q ℕ (p, q, t > 0). The same holds for the divisor of r_{0}, which is simply three times that of w_{0}. That leaves only the J-invariance, which as usual is the easiest part to check. Indeed, from (3.135) we get :

RJ = W^{−1} R W  (3.140)

Remark: Although we have postponed dealing with twins with equal tangency orders (p = q) until the “systematic” investigation of §5, it may be noted that in the above Example 11 (as also in Example 15 below) the action of J and J^{2} exchanges both types of twins (i.e. p ≠ q and p = q).

Example 12 . Invariance under K_{1} : (A, B) ↦ (B^{−1}, A). If we set successively :

W := \{\overline{\overline{B}}^{2} \overline{\overline{A}}^{3} B, \overline{\overline{A}}^{2} \overline{\overline{B}}^{3} A\}  (3.141)
Q := \{\overline{\overline{A}}^{3} B, \overline{\overline{B}}^{3} A\}  (3.142)
W_{1} := {Q, W }  (3.143)
R := W_{1}, W K_{1}_{1} W K_{1}^{2}_{1} W K_{1}^{3}_{1}  (3.144)

then the K_{1}-invariant relation R(F, G) ≡ 1 has a general twin-solution that depends on p/q and c.

Proof : Since K_{1}^{4} = 1, (3.144) gives R^{K^{1}} ≡ W _{1}^{−1} RW_{1} which takes cares of the invariance. Up to the innocuous factor :

T_{6 p+6 q+t,4 p+4 q} T_{p,q,q,q} T_{q,p,p,p} ≡ −(2 p + 2 q + t) p q (p − q)^{2} (p + q)^{2} (3.145)

the divisor associated with W_{1} coincides with the divisor of W, which we already encountered in Example 6 and found (see (3.114)) to be non-vanishing.

The reason for bracketing W with Q is of course a question of parity : whereas W and W^{K^{1}} have opposite corner components (since these are of even degree 6 in both a and b, and so change signs under (a, b) ↦ (−b, a)), the new words W_{1} and W K_{1}_{1} have identical corner components :

W K_{1}_{0} = −W_{0} ; qK_{1}_{0} = −q_{0} ; W K_{1}_{10} = W_{10}  (3.146)

so that the corner components (and divisors) of all four factors in (3.144) merely add up.

Example 13 . Invariance under Aut_{1} = {I, I_{0}, I_{1}, I_{2}}. Take any twin-generating relation W = {U, V } as in Example 5 or 6 and set W_{1} := {B, {A, W }}. Then the Aut_{1}-invariant relation :

R := W_{1} (W I_{1}_{1})^{−1} W I_{0}_{1} (W I_{2}_{1})^{−1}  (3.147)

has a general twin solution that depends on p/q and c.

Proof : The invariance follows from the relations :

R^{I^{0}} ≡ Q_{I_{0}} R QI_{0}^{−1} with Q_{I_{0}} = W_{1} (W I_{1}_{1})^{−1}  (3.148)
R^{I^{1}} ≡ Q_{I_{1}} R^{−1} QI_{1}^{−1} with Q_{I_{1}} = W I_{2}_{1} (W I_{0}_{1})^{−1}  (3.149)
R^{I^{2}} ≡ R^{−1}  (3.150)

On the other hand, W automatically has alternance (even,even) in (A,B), but W_{1} has alternance (odd,odd), and a divisor which differs from that of W by the trivial factor T_{2 d p+2 d q+t,p,q} (d being the global alternance of U or V ). As a consequence, all four factors in (3.147) contribute the same corner component and the same divisor.

Example 14 . Invariance under Aut_{2} = {I, I_{0}, S_{1}, S_{2}}. Take any U = U(A, B) that has symmetrical alternance (d_{1}, d_{2}), d_{1} = d_{2}, and yields a twin-generating relation W := {U, U^{S^{1}}}. For instance, take

U = \overline{\overline{A}}^{2} \overline{\overline{B}}^{3}A as in (3.113). Then the Aut_{2}-invariant relation R := W^{I^{0}} W = 1 has a general twin-solution that depends on p/q and c.

Proof : The invariance follows from :

R^{S^{1}} = W R^{−1} W^{−1} ; R^{S^{2}} = R^{−1} ; R^{I^{0}} = W R W^{−1}  (3.151)

Moreover, since W has an even (global) alternance 2 (d_{1} + d_{2}) = 4 d_{1} = 4 d_{2}, the two factors W and W^{I^{0}} which make up R contribute the same corner component and the same (non-vanishing) divisor – namely (3.114) if we take U as in (3.113).

Example 15 . Invariance under Aut_{3} = {I, J, J^{2}, S_{1}, S_{1} J, J S_{1}}. We take the same word W of alternance (10, 10) as in Example 11, but this time we set:

W_{1} = {{A, B}, W }  (3.152)
W_{2} = W_{1} (W_{1}^{S^{1}})^{−1} = {{A, B}, W } {W^{S^{1}}, {B, A}}  (3.153)
R = W_{2} W J_{2} W J^{2}_{2}  (3.154)

Then the Aut_{3}-invariant relation R = 1 has a general twin-solution that depends on p/q and c.

Proof : The whole point of bracketing W with {A, B} is of course to get a word W_{1} with the same divisor as W (upto the trivial factor T_{q,p} T_{10 p+10 q+t,p+q}) but such that W_{1} and (W_{1}^{S^{1}})^{−1} have the same corner components (rather than opposite ones as in the case of W). This yields a W_{2} verifying W S_{1}_{2} = W _{2}^{−1} and with twice the corner component of W_{1}. Moreover since the corner components of {A, B} and W are invariant under the action (3.136) of J, so too are those of W_{1} and W_{2}. The upshot is that the corner component and divisor of R are exactly six times those of W_{1}, which in turn are essentially the same as those of W.

As for the invariance, it is enough to check it for two generators of Aut_{3}, e.g. J and S_{1}. Due to W S_{1}_{2} = W _{2}^{−1}, the definition of R yields :

RJ = W _{2}^{−1} R W_{2} ; R^{S^{1}} = W _{2}^{−1} R^{−1} W_{2}  (3.155)

Example 16 .Invariance under Aut_{4} = {I, I_{0}, I_{1}, I_{2}, S_{1}, S_{2}, K_{1}, K_{2}}. If we take a word W = W (A, B) such that :

(*) W be twin-generating (**) W^{S^{1}} = W^{−1}

(***) W be of alternance (d_{1}, d_{2}) in (A,B) with d_{1} = d_{2} = odd

and if we set :

R := W W^{K^{1}} W^{I^{0}} W^{K^{2}} = W W^{K^{1}} W^{K{2 \atop 1}} W^{K{3 \atop 1}}  (3.156)

then the Aut_{4}-invariant relation R = 1 has a general twin solution that depends on p/q and c. For instance, a suitable choice for W is given by the following steps :

W_{1} = \{U_{1}, US_{1}_{1}\} and U_{1} = \overline{\overline{A}}^{2} \overline{\overline{B}}^{3} A  (3.157)
W_{2} = {U_{2}, US_{1}_{2}} and U_{2} = A  (3.158)
W_{3} = \{U_{3}, US_{1}_{3}\} and U_{3} = \overline{\overline{B}}^{3} A  (3.159)
W_{4} = {W_{3}{W_{2}, W_{1}}}  (3.160)
W = W_{4} (W S_{1}_{4})^{−1}  (3.161)

Proof : As soon as W verifies (*) and (**), the relation R defined by (3.156) verifies :

R^{K^{1}} = W^{−1} R W ; R^{S^{1}} = W^{−1} R^{−1} W  (3.162)

which ensures the invariance, since K_{1} and S_{1} generate Aut_{4}. Moreover, the imparity condition (***) makes sure that the corner components of the four factors in (3.156) add up rather than self-destruct.

But the existence of words W with these properties is far from obvious. So let us check the suitability of the particular W mentioned above. We observe that the divisor D_{1}(p, q; t) of W_{1} is given by (3.114) and that the divisor D(p, q; t) of W is essentially the same :

D(p, q; t) = 2 p q (p − q)^{4} (p + q)^{2} (3 p + 3 q + t) (3 p + 5 q + t) D_{1}(p, q; t) (3.163)

with an innocuous front factor which merely reflects the bracketing of W_{1} by W_{2} and W_{3}, and is therefore equal to :

T_{6 p+6 q+t,p+q,4 p+4 q} T_{p,q,q,q} T_{q,p,p,p} T_{q,p} T_{3 p+q,3 q+p}  (3.164)

It is natural that W_{2} and W_{3} should contribute only trivial factors, since neither U_{2} nor U_{3} have equal alternance in A and B, so that for j = 1, 2 the corner component of {U_{j}, U_{j}^{S_{1}}} cannot vanish identically under all realisations (a, b) ↦ (l_{p}, l_{q}).

Despite its forbidding complexity, this example is the ‘simplest of its kind’, at least as long as we insist on working with multicommutators and demand a certain symmetry in A and B. But if we drop these requirements, we may produce a slightly simpler example :

Example 17 . Invariance under Aut_{4} : simpler example. If we set :

U := \{(\overline{\overline{A}}^{3} B \{(\overline{\overline{A}} B), B^{−1}\}\}  (3.165)
V := \{(\overline{\overline{A}}^{2} B \{(\overline{\overline{A}}^{2} B), \overline{\overline{A}}(B^{2})\}\}  (3.166)
W_{1} := U V  (3.167)
W_{2} := \overline{\overline{B}}^{2} W_{1}  (3.168)
W := W_{2} (W S_{1}_{2})^{−1}  (3.169)
R := W W^{K^{1}} W^{I_{0}} W^{K^{2}}  (3.170)

then the Aut_{4}-invariant relation R = 1 has a general twin solution that depends on p/q and c.

Proof : It is enough to check that the new W still meets all three conditions (*) (**) (***) of Example 16. Clearly, if we set b_{n} := \overline{a}^{n} b, we find for the corner components of U, V , W_{1}, W_{2} :

u_{0} = −\overline{b}_{3} \overline{b}_{2} b ; v_{0} = 2 \overline{b}_{2} \overline{b}_{2} b_{1}  (3.171)
w_{10} = u_{0} + v_{0} ; w_{20} = \overline{b} \overline{b} (u_{0} + v_{0})  (3.172)

Next, we check that w_{10} (and so w_{20}) vanish under all realisations (a, b) ↦ (l_{p}, l_{q}). Then we calculate the divisor attached to W_{1} :

D_{1}(p, q; t) = −(p − q)^{2} q t^{2} (t − p + q)  (3.173)

and the one attached to W_{2} :

D_{2}(p, q; t) ≡ T_{3 p+5 q+t,q,q} D_{1}(p, q; t)
≡ (t + 3 p + 4 q) (t + 3 p + 5 q) D_{1}(p, q; t)  (3.174)

From that we infer the divisor attached to W = W_{2} (W S_{1}_{2})^{−1} (beware of the + sign) :

D(p, q; t) = D_{2}(p, q; t) + D_{2}(q, p; t) = −(p − q)^{2} t^{2} Δ(p, q; t)  (3.175)

with

Δ(p, q; t) = (p + q) t^{3} + (10 p^{2} + 10 p q + 10 q^{2}) t^{2}  (3.176)

+(29 p^{3} + 27 p^{2} q + 27 p q^{2} + 29 q^{3}) t + (p − q)^{2} (20 p^{2} + 38 p q + 20 q^{2})

Therefore neither Δ nor D vanish for p ≠ q and t in p ℕ+q ℕ (p, q, t > 0). So W verifies (*) and (**). It also verifies (***), since it clearly has alternance (5, 5) in (A,B).

3.6 Further examples

Example 18 Several unmovable parameters : Let W_{0} := W = { U, V } be a twin-generating relation like in Examples 5 or 6, with alternance (m_{0}, n_{0}) in (A, B), and let W_{j}, (j = 1, ⋯, l) be l non-twin-generating relations with alternance (m_{j}, n_{j}), for instance :

W_{j} := \overline{\overline{A}}^{m_{j}−1} \overline{\overline{B}}^{n_{j}} A  (3.177)

Further, assume that m_{j}, n_{j} increase so fast that the integers μ_{j}, ν_{j} derived therefrom :

μ_{j} := m_{j} − (m_{0} + m_{1} + ⋯ m_{j−1})  (3.178)
ν_{j} := n_{j} − (n_{0} + n_{1} + ⋯ n_{j−1}) (j = 1, 2, ⋯ l)

are themselves positive and strictly increasing. Then, if we set :

R_{0} := W_{0} ; R_{j} := {W_{j}, R_{j−1}} (j = 1, ⋯, l)  (3.179)

the relation R_{l}(F, G) = 1 has a general twin solution which depends not only on the usual parameters p/q and c = α^{q_{∗}} β^{−p_{∗}}, but also on l new parameters c_{1}, ⋯, c_{l} which may assume any complex value, but are ‘unmovable’ in the sense that, for each of them, the place of ‘first occurence’ in F or G is within coefficients of a well-defined depth, namely t = t_{j} := μ_{j} p + ν_{j} q for c_{j}.

Proof : The divisor D_{l}(p, q; t) of R_{l} is equal to the divisor D(p, q; t) of W multiplied :

(i) by the trivial (i.e. t-independent) factors T_{p,q^{(n_{j})},p^{(m_{j} −1)}} contributed by the words W_{j} (j = 1, ⋯, l).

(ii) by the elementary (i.e. t-affine) factors (t − μ_{j} p − ν_{j} q) (j = 1, ⋯, l) which we introduced by bracketing R_{j−1} with W_{j}.

Therefore D_{l}(p, q; t_{j}) ≡ 0 for t_{j} := μ_{j} p+ν_{j} q (j = 1, ⋯, l). Under normal circumstances, this should prevent the existence of twin-solutions, but here it has the opposite effect of enlarging their number, by introducing l new free parameters c_{j}. Indeed, for any initial conditions (p, q) and (α, β), the relation R_{l}(F, G) ≡ 1 is equivalent to the system :

log R_{j−1} ≡ c_{j} log W_{j} (mod. terms of depth t_{j+1}) (j = 1, ⋯, l) (3.180)

which, for any given choice (c_{1}, ⋯, c_{l}) in ℂ^{l}, clearly admits (upto conjugacy) a unique solution (F, G).

The coefficients of F, G with depth t < t_{1} are calculated inductively exactly as if we were dealing with the sole relation R_{0}(F, G) := W (F, G) = 1. Then the coefficients with depth t_{1} ≤ t ≤ t_{2} are calculated from log R_{0} = c_{1} log W_{1}; those with depth t_{2} ≤ t ≤ t_{3} from log R_{1} = c_{2} log W_{2}; etc ...; and lastly those of depth t_{l} ≤ t from log R_{l−1} = c_{l} log W_{l}.

Actually, the conclusion would remain unchanged if the (μ_{j}, ν_{j}), instead of forming an increasing sequence, were pairwise distinct and comparable (for the natural order on ℕ^{2}), while of course remaining positive. For noncomparable pairs, however, there would occur some slight changes, since the order of the sequence formed by the zeros t_{j} of the divisor’s ‘elementary factors’ would depend on the tangency ratio p/q.

Example 19 . One movable parameter : Let R_{1} and R_{2} be two twin-generating relations of the type encountered in Example 9 (see (3. 120-121)), with invariance under I_{0}:

R_{j}(A^{−1}, B^{−1}) = (R_{j}(A, B))^{−1} (j = 1, 2)  (3.181)

Then if we set :

R(A, B) := R_{2}(A, R_{1}(AB))  (3.182)

the relation R(F, G) = 1 has a general twin-solution which, on top of the usual parameters p/q and c = α^{q_{∗}} β^{−p_{∗}}, depends on two new parameters : one continuous parameter c_{1}, which may assume any complex value, and one discrete parameter r_{1}, which denotes the place (or ‘depth’) of first occurence of c_{1}, and may assume any entire value.

Proof : Fix three pairwise distinct integers p, q, r_{1}. Then set p_{2} := p, q_{2} := m_{1}p+n_{1}q+r_{1}, where (m_{1}, n_{1}) denotes the alternance of R_{1}(A, B) in (A, B). Upto conjugacy, the general solution of R_{1}(F, G) = 1 may be written in normal-conormal form (see (3.9) and (3.10)) as :

F = F^{nor} = α x^{p+1}∂ ; G = G^{conor} = β (x^{q+1} + ⋯) ∂  (3.183)

Similarly, the general solution of R_{2}(F, K) = 1 may be written as :

F = F^{nor} = α x^{p_{2}+1}∂ ; K = K^{conor} = γ (x^{q_{2}+1} + ⋯) ∂  (3.184)

with K^{conor} depending only on (p_{2}, q_{2}) := (p, m_{1}p + n_{1}q + r_{1}) and (α, β), so that F^{nor} is the same in (3.183) and (3.184).

Example 20 . Twin glueing. Let R_{1}(A, B) ..., R_{r}(A, B) be distinct twin-generating relations of the type encountered before, and let R_{0}(A, B) := \overline{\overline{A}}^{n_{0}} \overline{\overline{B}}^{m_{0}}A be a multicommutator of very high alternance (n_{0}, m_{0}). Then for each i = 1, …, r, the relation :

R(A, B) := R_{0}(A, B) {..{R_{1}(A, B), R_{2}(A, B)} … R_{r}(A, B)} (3.185)

has a twin-solution which coincides upto high order (but not exactly) with the twin solution of R_{i}(A, B).

Proof : This is simply because for each i the corresponding divisor coincides, upto a non-vanishing factor, with the divisor D_{i}(t) associated with the isolated equation R_{i}(A, B).

The separateness of the i solutions (each with its own parameters) stands in sharp contrast to the connectednes of the solutions of what would be the differential equation analogue :

\underline{r}(a, b) := \underline{r}_{0}(a, b) + [..[\underline{r}_{1}(a, b), \underline{r}_{2}(a, b)] … \underline{r}_{r}(a, b)]  (3.186)

with \underline{r}_{i}(a, b) denoting the lowest-degree homogeneous component in the series r_{i}(a, b) := log R_{i}(e^{a}, e^{b}) and with a = φ ∂ , b = ψ ∂ for, say, φ given and ψ unknown. Here too, each ‘factor’ \underline{r}_{i}(a, b) contributes its own parameters, but the multibracket on the right-hand side of (3.188) introduces r − 1 additional parameters which have the effect of connecting (under continuous deformations) the separate solutions.

Thus, whereas the solutions of differential equations may be seamlessly ‘welded’ together, the solutions of composition equations can only be ‘glued’^{4}. This reflects a very basic difference between the two classes of problems. Nor is it due to the fact that we are solving our (differential or composition) equations in rings of powers series : the difference persists, undiminished, when we go over to transserial solutions (see §8 infra).

4 Some Lie theory. Active/passive subalgebras. Divisors and universal kernels.

4.1 The active/passive filtration of Lie[a, b].

The twenty examples of twins reviewed in the previous section are easy to construct but somewhat atypical in so far as all of them verify relations made up of suitably arranged multi-commutators. Before turning to the description of truly ‘generic’ twins, we must insert a section devoted to the natural filtrations that arise on free algebras when they get represented as one-variable differential algebras.

Let us for simplicity deal with the two-generator algebra Lie[a, b]. It admits a natural sequence of decreasing ideals ^{5} :

Lie[a, b] = Uker_{0} ⊃ Uker_{1} ⊃ Uker_{2} ⊃ … Uker_{∞} = Pass (4.187)
[Uker_{i}, Uker_{j}] ⊂ Uker_{i+j}  ∀i, j (0 ≤ i, j ≤ ∞) (4.188)

The k-th ideal Uker_{k} is defined as consisting of all w(a, b) ∈ Lie[a, b] which vanish upto order k:

{w ∈ Uker_{k}}  ⟺  {w(F_{∗}, G_{∗}) = O(ϵ^{k}) ∀p, q, φ, ψ} (4.189)

4with only a week interaction stemming from the perturbation R_{0}.

5with Uker standing for universal kernel.

whenever the pair (a, b) gets replaced by a pair (F_{∗}, G_{∗}) of the form :

F_{∗} := (x^{p} + ϵ φ(x)) x ∂_{x} , G_{∗} := (x^{q} + ϵ ψ(x)) x ∂_{x}  (4.190)

or, what amounts to the same^{6}, of either form :

\begin{aligned}F_{∗} & := & ∂_{z} & , & G_{∗} & := & (z^{−q} + ϵ Ψ(z)) z ∂_{z} \\ F_{∗} & := & (z^{−p} + ϵ Φ(z)) z ∂_{z} & , & G_{∗} & := & ∂_{z} \end{aligned}

Remark 1 : The reason for limiting ourselves to pairs (F_{∗}, G_{∗}) consisting of perturbed monomials is of course that we are mostly concerned with twins in the ring of power series. But even when investigating transserial twins (see §8), the ordinary monomials retain their primacy, due to the fact that they alone, of all transmonomials, enjoy a double stability : under multiplication and composition.

The ideal Uker_{0} coincides with the whole algebra Lie[a, b], while Uker_{∞} consists of all elements w that vanish identically in all one-variable differential representations of Lie[a, b]. For our present purposes they are as if nonexistent, and deserve to be regarded as constituting the ‘passive’ subalgebra Pass. What matters is the ‘active’ algebra Act, ie the quotient of Lie[a, b] by Pass.

Actually, we shall require the full filtration :

Act ≃ Act_{0} ⊕ Act_{1} ⊕ Act_{2} … with Act_{i} = Uker_{i}/Uker_{i+1}(4.191)
Act ≃ Act_{0} ⊕ Act_{∗}  with Act_{∗} = Uker_{1}/Uker_{∞}(4.192)

Of special importance is the leading active algebra Act_{0} := Lie/Uker_{1}. Its (homogeneous) elements w(a, b) vanish in all ‘monomial’ representations, but possess non-vanishing divisors D(t) := D_{w}(t; p, q).^{7}

Remark 2 : from filtration to graduation. In order to turn the natural filtration (4.187) into a graduation of type (4.191), one would have to actually embed each Act_{i} into Lie[a, b]. The most natural way of accomplishing this is by ruling that all the embedded components Act_{i} should be orthogonal to one another and to Pass , with respect to the natural scalar product on Lie[a, b]^{8}. This again would lead to worthwhile developments which, however, would be a distraction from our present investigation.

6under a suitable variable change x = h_{ϵ}(z)

7We have already encountered many instances of divisors. For a general definition, valid even in the case of more than two generators a, b, c …, see §6.6.

Next comes the question of the dimensions, which makes sense only inside specified homogeneous components Lie[a, b]^{(d_{1},d_{2})}.

For any given degree (d_{1}, d_{2}), the components Uker_{i}^{(d_{1},d_{2})} clearly become stationary after a certain critical i := i_{crit}(d_{1}, d_{2}). This stationary ideal component coincides with Uker_{∞}^{(d_{1},d_{2})}. Of course the corresponding components Act_{i}^{(d_{1},d_{2})} of the active algebra turn empty.

For any given index i, the dimensions dim(Uker_{i}^{(d_{1},d_{2})}) and also, less obviously, dim(Act_{i}^{(d_{1},d_{2})}), are non-decreasing (but non-convex) functions of d_{1} and d_{2}.

Remark 3 : The spaces [Uker_{i′}^{(d′_{1},d′_{2})}, Uker_{i′′}^{(d′′_{1},d′′_{2})}] are clearly (strict) subspaces of Uker_{i′+i′′}^{(d′_{1}+d′′_{1},d′_{2}+d′′_{2})}. Taking into account the corresponding quotients^{9} Act_{(i′,i′′)}^{(d′_{1},d′′_{1}),(d′_{2},d′′_{2})} would lead to a considerable refinement of our active/passive filtration, but we need not go into that here.

Before establishing the exact formulas for the main dimensions, we adduce two tables.

The first table extends to all degrees (d_{1}, d_{2}) ≤ (15, 15) and its entries are three-number columns :

top  : α = dim(Act_{0}^{(d_{1},d_{2})})
middle : β = dim(Act_{∗}^{(d_{1},d_{2})})
bottom : γ = dim(Pass_{0}^{(d_{1},d_{2})})

When Act_{∗}^{(d_{1},d_{2})} doesn’t reduce to its first component Act_{1}^{(d_{1},d_{2})}, the number β is entered as ^{∗}β.

The second table extends to all degrees (4, 4) ≤ (d_{1}, d_{2}) ≤ (14, 14) and gives the exact decomposition β = β_{1} + β_{2} + β_{3} + …:

8ie the one induced by the natural scalar product on the envelopping free associative algebra Ass(a, b)

9which may be joined to form Lie algebras.

top : β = dim(Act_{∗}^{(d_{1},d_{2})}) written as ^{∗}β when β ≠ β_{1}
lower : β_{1} = dim(Act_{1}^{(d_{1},d_{2})})
lower : β_{2} = dim(Act_{2}^{(d_{1},d_{2})})
lower : β_{3} = dim(Act_{3}^{(d_{1},d_{2})})

…

Actually, the calculations are rather costly and we did not reach the point where β_{3} becomes > 0, although this certainly takes place within the limits of the table. The uncalculated numbers have been replaced by dots.^{10}

10Entries for d_{1} or d_{2} < 4 were left out since for them β = β_{1} and 0 = β_{2} = β_{3} . . . ..

1 2 3 4  5  6  7  8  9  10
1 1 1 1 1  1  1  1  1  1  1
0 0 0 0  0  0  0  0  0  0
0 0 0 0  0  0  0  0  0  0
2 1 1 2 2  3  3  4  4  5  5
0 0 0 0  0  0  0  0  0  0
0 0 0 0  0  0  0  0  0  0
3 1 2 3 4  4  5  6  6  7  8
0 0 0 1  2  3  4  6  7  9
0 0 0 0  1  1  2  3  4  5
4 1 2 4 5  6  6  7  8  9  9
0 0 1 2  5  8 12  15  21  26
0 0 0 1  3  6 11  17  25  35
5 1 3 4 6  7  8  8  9  10  11
0 0 2 5  9 15 22 ^{∗}31 ^{∗}40  ^{∗}52
0 0 1 3  9 19 36  59  93  137
6 1 3 5 6  8  9 10  10  11  12
0 0 3 8 15 ^{∗}23 ^{∗}36 ^{∗}49 ^{∗}67  ^{∗}86
0 0 1 6 19 43 86 153 255  399
7 1 4 6 7  8 10 11  12  12  13
0 0 4 12 22 ^{∗}36 ^{∗}52 ^{∗}74 ^{∗}100 ^{∗}132
0 0 2 11 36 86 182 343 603  999
8 1 4 6 8  9 10 12  13  14  14
0 0 6 15 ^{∗}31 ^{∗}49 ^{∗}74 ^{∗}103 ^{∗}142 ^{∗}186
0 0 3 17 59 153 343 684 1274 2224
9 1 5 7 9 10 11 12  14  15  16
0 0 7 21 ^{∗}40 ^{∗}67 ^{∗}100 ^{∗}142 ^{∗}192 ^{∗}257
0 0 4 25 93 255 603 1274 2493 4589
10 1 5 8 9 11 12 13  14  16  17
0 0 9 26 ^{∗}52 ^{∗}86 ^{∗}132 ^{∗}186 ^{∗}257 ^{∗}339
0 0 5 35 137 399 999 2224 4589 8869
11 1 6 8 10 12 13 14  15  16  18
0 0 11 33 ^{∗}65 ^{∗}111 ^{∗}168 ^{∗}243 ^{∗}332 ^{∗}445
0 0 7 48 196 604 1586 3720 8050 16333
12 1 6 9 11 12 14 15  16  17  18
0 0 13 39 ^{∗}82 ^{∗}137 ^{∗}214 ^{∗}306 ^{∗}426 ^{∗}568
0 0 8 62 270 875 2423 5966 13552 28786
13 1 7 10 12 13 15 16  17  18  19
0 0 15 ^{∗}48 ^{∗}99 ^{∗}172 ^{∗}265 ^{∗}387 ^{∗}535 ^{∗}721
0 0 10 80 364 1241 3595 9286 22057 49002
14 1 7 10 12 14 15 17  18  19  20
0 0 18 ^{∗}56 ^{∗}120 ^{∗}208 ^{∗}328 ^{∗}477 ^{∗}668 ^{∗}897
0 0 12 100 478 1709 5192 14025 34843 80769
15 1 8 11 13 15 16 18  19  20  21
0 0 20 ^{∗}67 ^{∗}142 ^{∗}254 ^{∗}399 ^{∗}589 ^{∗}822 ^{∗}1116
0 0 14 124 618 2313 7335 20710 53635 129613

51

4  5  6  7  8  9 10  11  12  13  14
4 2  5  8 12 15 21 26  33  39 ^{∗}48 ^{∗}56
2  5  8 12 15 21 26  33  39  47  55
0  0  0  0  0  0  0  0  0  1  1
5 5  9 15 22 ^{∗}31 ^{∗}40 ^{∗}52 ^{∗}65 ^{∗}82 ^{∗}90 ^{∗}120
5  9 15 22 29 35 43  52  63  72  83
0  0  0  0  2  5  9  13  19  27  37
6 8 15 ^{∗}23 ^{∗}36 ^{∗}49 ^{∗}67 ^{∗}86 ^{∗}111 ^{∗}137 ^{∗}172 ^{∗}208
8 20 21 30 38 47 56  66  76 …  …
0  0  2  6 11 . . . . . .  …  …  …  …
7 12 22 ^{∗}36 ^{∗}52 ^{∗}74 ^{∗}100 ^{∗}132 ^{∗}168 ^{∗}214 ^{∗}265 ^{∗}328
12 22 30 37 48 . . . . . .  …  …  …  …
0  0  6 . . . . . . . . . . . .  …  …  …  …
8 15 ^{∗}31 ^{∗}49 ^{∗}74 ^{∗}103 ^{∗}142 ^{∗}186 ^{∗}243 ^{∗}306 ^{∗}387 ^{∗}477
15 29 38 48 . . . . . . . . .  …  …  …  …
0  2 11 . . . . . . . . . . . .  …  …  …  …
9 21 ^{∗}40 ^{∗}67 ^{∗}100 ^{∗}142 ^{∗}192 ^{∗}257 ^{∗}332 ^{∗}426 ^{∗}535 ^{∗}668
21 35 47 . . . . . . . . . . . .  …  …  …  …
0  5 . . . . . . . . . . . . . . .  …  …  …  …
10 26 ^{∗}52 ^{∗}86 ^{∗}132 ^{∗}186 ^{∗}257 ^{∗}339 ^{∗}445 ^{∗}568 ^{∗}721 ^{∗}897
26 43 56 . . . . . . . . . . . .  …  …  …  …
0  9 . . . . . . . . . . . . . . .  …  …  …  …
11 33 ^{∗}65 ^{∗}111 ^{∗}168 ^{∗}243 ^{∗}332 ^{∗}445 ^{∗}579 ^{∗}746 ^{∗}943 ^{∗}1185
33 52 66 . . . . . . . . . . . .  …  …  …  …
0 13 . . . . . . . . . . . . . . .  …  …  …  …
12 39 ^{∗}82 ^{∗}137 ^{∗}214 ^{∗}306 ^{∗}426 ^{∗}568 ^{∗}746 ^{∗}956 ^{∗}1219 ^{∗}1525
39 63 76 . . . . . . . . . . . .  …  …  …  …
0 19 . . . . . . . . . . . . . . .  …  …  …  …
13 ^{∗}48 ^{∗}99 ^{∗}172 ^{∗}265 ^{∗}387 ^{∗}535 ^{∗}721 ^{∗}943 ^{∗}1219 ^{∗}1545 ^{∗}1945
47 72 . . . . . . . . . . . . . . .  …  …  …  …
1 27 . . . . . . . . . . . . . . .  …  …  …  …
14 ^{∗}56 ^{∗}120 ^{∗}208 ^{∗}328 ^{∗}477 ^{∗}668 ^{∗}897 ^{∗}1185 ^{∗}1525 ^{∗}1945 ^{∗}2441
55 83 . . . . . . . . . . . . . . .  …  …  …  …
1 37 . . . . . . . . . . . . . . .  …  …  …  …

4.2 Dimensions.

Throughout this section, we shall use the following notations :

p(n) := nb of partitions of n with positive summands  (4.193)
p^{∗}(n) := 1 + p(1) + p(2) + ⋯ + p(n)  (4.194)
p(n, m) := nb of partitions of n with m non-negative summands (4.195)
:= nb of partitions of n with at most m positive summands (4.196)
P (n, m) := nb of partitions of n with m positive summands  (4.197)

Clearly, p (•, •) and P (•, •) are expressible in terms of each other ;

p(n, m) ≡ P (n + m, m) ; P (n, m) ≡ p(n, m) − p(n, m − 1) (4.198)

Proposition 4.1 Full algebra : The dimensions L^{(d_{1},d_{2})} := dim Lie[a, b]^{(d_{1},d_{2})} are given by the classical formula:

L^{(d_{1},d_{2})} = \frac{1}{d_{1} + d_{2}} ∑_{δ|d_{1} , δ|d_{2}} μ(δ) \frac{((d_{1} + d_{2})/δ)!}{(d_{1}/d)! (d_{2}/d)!}  (4.199)

with μ(.) as the Möbius function.

Proposition 4.2 Full active algebra : The dimensions D^{(d_{1},d_{2})} := dim Act^{(d_{1},d_{2})} = codim Uker_{∞}^{(d_{1},d_{2})} are given by :

D^{(d_{1},d_{2})} = p(d_{1} + d_{2} − 1, d_{1}) + p(d_{1} + d_{2} − 1, d_{2}) − p(d_{1}+d_{2}−1) (4.200)
= P (2d_{1} + d_{2} − 1, d_{1}) + P (d_{1} + 2d_{2} − 1, d_{2}) − p(d_{1}+d_{2}−1)(4.201)
= p(d_{1} + d_{2} − 1, d_{1}) − p^{∗}(d_{1} − 1)  (if d_{2} − d_{1} ≥ −3) (4.202)
= p(d_{1} + d_{2} − 1, d_{2}) − p^{∗}(d_{2} − 1)  (if d_{1} − d_{2} ≥ −3) (4.203)

Proposition 4.3 Leading active algebra : The dimensions D_{0}^{(d_{1},d_{2})} := dim Act_{0}^{(d_{1},d_{2})} = codim Uker_{0}^{(d_{1},d_{2})} are given by :

D_{0}^{(d_{1},d_{2})} = 1 + E(\frac{(d_{1} − 1)(d_{2} − 1))}{d_{1}}) + E(\frac{(d_{1} − 1)(d_{2} − 1))}{d_{2}})  (4.204)

where E(x) denotes the entire part of the real number x.

Proposition 4.4 Secondary active algebras : For any fixed index i ≥ 0 and any fixed d_{1} (resp d_{2}) ≥ sup(3, i + 1), the dimensions D_{i}^{(d_{1},d_{2})} := dim Act_{i}^{(d_{1},d_{2})} = dim Uker_{i}^{(d_{1},d_{2})} − dim Uker_{i+1}^{(d_{1},d_{2})} grow roughly like d_{2}^{i+1} (resp d_{1}^{i+1}) times some constant.

More precisely, though none of the generating functions ∑_{d_{1},d_{2}} D_{i}^{(d_{1},d_{2})} x_{1}^{d_{1}} x_{2}^{d_{2}} is rational, the partial sums ∑_{d} D_{i}^{(d_{0},d)} x^{d} = ∑_{d} D_{i}^{(d,d_{0})} x^{d} are all rational, with denominators apparently of the form (1 − x^{i+1})(1 − x^{(i+1) d_{0}}).

4.3 Universal constraints.

A non-increasing sequence \underline{ν} := (n_{1}, n_{2}, …, n_{s}) with n_{s} ≥ 1 and n = ∑ n_{i} is said to be a positive partition of n. We write for short \underline{ν} > 0 and ∥\underline{ν}∥ = n. A non-increasing sequence \overline{μ} := (m_{1}, m_{2}, …, m_{s}) with m_{s} ≥ 0 and m = ∑ m_{i} is said to be a non-negative partition of m. We write for short \overline{μ} ≥ 0 and ∥\overline{μ}∥ = m.

Let ψ be any smooth one-variable function with algebraically independent derivatives ψ, ψ′, ψ′′, ψ′′′ … For any non-negative partition \overline{μ} we set :

ψ\overline{μ}(x) = ψ\overline{(m_{1},...,m_{r})}(x) := ψ^{(m_{1})}(x) … ψ^{(m_{r})}(x)  (4.205)

We denote by ℍ (resp ℍ_{d_{1}}^{d_{1}+d_{2}−1}) the Lie algebra over ℂ spanned by all operators ψ\overline{μ}(x) ∂_{x} (resp those operators whose index \overline{μ} is a non-negative partition of the number d_{1} + d_{2} − 1 into d_{1}-summands).

Proposition 4.5 Representation of the active algebra Act : The component Act^{d_{1},d_{2}} of the quotient

Act := Lie[a, b]/Uker_{∞}  (4.206)

is isomorphic to the subspace of ℍ_{d_{1}}^{d_{1}+d_{2}−1} that is orthogonal to the ‘Lagrangian constraints’ Lag\underline{ν} or to the ‘power constraints’ Pow\underline{ν} with indices of the form \underline{ν} = \underline{∅} or of the form \underline{ν} = \underline{(n_{1}, …, n_{s})} with

∑ n_{i} ≤ d_{1} − 2  and  s + ∑ n_{i} ≤ d_{1} + d_{2} − 1  (4.207)

In symmetric fashion, it is also isomorphic to the subspace of ℍ_{d_{1}}^{d_{1}+d_{2}−1} that is orthogonal to the constraints Lag\underline{ν} or Pow\underline{ν} with indices of the form \underline{ν} = \underline{∅} or of the form \underline{ν} = \underline{(n_{1}, …, n_{s})} with

∑ n_{i} ≤ d_{2} − 2  and  s + ∑ n_{i} ≤ d_{1} + d_{2} − 1  (4.208)

By linearity, the constraints Lag\underline{ν} and P ow\underline{ν} are wholly determined by the

‘constraint tensors’ Lag\overline{μ} ; \underline{ν} and P ow\overline{μ} ; \underline{ν} such that :

< Lag\underline{ν} , ψ\overline{μ} > = Lag\overline{μ} ; \underline{ν}  (4.209)
< Pow\underline{ν} , ψ\overline{μ} > = Pow\overline{μ} ; \underline{ν}  (4.210)

4.4 Lagrangian constraints.

Proposition 4.6 Tensor of ‘Lagrangian constraints’ : It is explicitely given by :

Lag\overline{μ} ; \underline{ν} = Lag\overline{(m_{1},...,m_{r})} ; \underline{(n_{1},...,n_{s})}  (4.211)
≡ ∑_{\underline{ν}=\underline{ν_{1}}⊕···⊕\underline{ν_{r}}} lag\underline{ν_{1}}(m_{1}) … lag\underline{ν_{r}}(m_{r})  (4.212)

with a sum extending to all decompositions \underline{ν} = \underline{ν_{1}} ⊕ ⋯ ⊕ \underline{ν_{r}} of \underline{ν} into r sub-partitions \underline{ν_{i}} (some of which may be empty) and with integers lag\underline{ν_{i}}(m_{i}) defined by :

lag\underline{∅}(m) := 1  (∀m ∈ ℕ)  (4.213)

and for non-empty partitions \underline{ν} by means of the identity :

(−1)^{m} ψ^{(m)}(y) := φ^{−1}(x) (φ′(x))^{m} + ∑_{1≤s; 1≤n_{i}}  (4.214)

lag\underline{(n_{1},...,n_{s})}(m) φ^{∥n∥−1} (φ′(x))^{m+s−∥n∥} ∏_{1≤i≤s}(φ)^{(1+n_{i})}(x)

that connects the successive derivatives of two functions φ(x) and ψ(y) linked by the reciprocity relation :

\{y = ∫ ^{x} \frac{dx_{1}}{φ(x_{1})}\} ⟺ \{x = ∫ ^{y} \frac{dy_{1}}{ψ(y_{1})}\}  (4.215)

Remark 1: In practical terms, the identities (4.214) are established – and the coefficients therein can be calculated – by successive differentiations of the identities :

ψ ∘ h (x) ≡ h′ (x) ≡ 1/φ(x)  with h(x) ≡ y  (4.216)

which are reminiscent of Lagrange’s inversion formula : hence the name given to the corresponding constraints.

Remark 2: An easy induction shows that for small values of m:

lag\underline{ν}(m) ≡ 0  if 0 ≤ m ≤ s − 1 + ∥ν∥  (4.217)

and that for larger values each lag\underline{ν}(m) resolves into an exponential^{11} sum :

lag\underline{ν}(m) = P\underline{ν}^{0}(m) + ∑_{1≤j≤∥ν∥} (−j)^{m} P\underline{ν}^{j}(m)  (4.218)

with coefficients P\underline{ν}^{j}(m) that are polynomial in m, with well-defined degrees. Actually, even for m in the interval 1 ≤ m ≤ s − 1 + ∥ν∥, the exponential sum (4.218) still yields the right answer, ie 0. For m = 0, however, it yields non-zero values, whereas the correct value is 0, as in definiton (4.217).

Remark 3: Expanding (4.212) for s = 2 and s = 3 we find :

Lag\overline{(m_{1},...,m_{r})} ; \underline{(n_{1},n_{2})} = ∑_{i} lag\underline{(n_{1},n_{2})}(m_{i}) + ∑_{i≠j} lag\underline{(n_{1})}(m_{i}) lag\underline{(n_{2})}(m_{j})  (4.219)
Lag\overline{(m_{1},...,m_{r})} ; \underline{(n_{1},n_{2},n_{3})} = ∑_{i} lag\underline{(n_{1},n_{2},n_{3})}(m_{i}) + ∑_{i≠j} lag\underline{(n_{1})}(m_{i}) lag\underline{(n_{2},n_{3})}(m_{j})

+ ∑_{i≠j} lag\underline{(n_{2})}(m_{i}) lag\underline{(n_{1},n_{3})}(m_{j}) + ∑_{i≠j} lag\underline{(n_{3})}(m_{i}) lag\underline{(n_{1},n_{2})}(m_{j})

+ ∑_{i≠j,j≠k,k≠i} lag\underline{(n_{1})}(m_{i}) lag\underline{(n_{2})}(m_{j}) lag\underline{(n_{3})}(m_{k})  (4.220)

11with respect to m.

Remark 4: Let us write down the exponential sums lag\underline{ν}(m) for the first seven partitions, ie for ∥\underline{ν}∥ ≤ 3 :

lag\underline{(∅)}(m) = 1
lag\underline{(1)}(m) = (1 − m)
lag\underline{(2)}(m) = (\frac{3}{4} − \frac{1}{2}m) + (−1)^{m} \frac{1}{4}
lag\underline{(1,1)}(m) = (\frac{7}{4} − 2m + \frac{1}{2}m^{2}) + (−1)^{m} \frac{1}{4}
lag\underline{(3)}(m) = (\frac{11}{36} − \frac{1}{6}m) + (−1)^{m} \frac{1}{4} + (−2)^{m} (−\frac{1}{18})
lag\underline{(2,1)}(m) = (−\frac{23}{9} + \frac{29}{12}m − \frac{1}{2}m^{2}) + (−1)^{m} (−1 + \frac{1}{4}m) + (−2)^{m} (\frac{1}{18})
lag\underline{(1,1,1)}(m) = (+\frac{13}{4} − \frac{49}{12}m + \frac{3}{2}m^{2} − \frac{1}{6}m^{3}) + (−1)^{m} (\frac{3}{4} − \frac{3}{4}m)

For m = 0 we must posit lag\underline{ν}(0) := 0, but for small, positive values of m (ie smaller than s − 1 + ∥ν∥) the above formulas, of their own, yield 0.

4.5 Power constraints.

Proposition 4.7 Tensor of ‘power constraints’ : It is explicitely given by Pow\overline{μ}_{\underline{∅}} = 1 and for non-empty partitions ν by :

Pow\overline{μ} ; \underline{ν} = Pow\overline{(m_{1},...,m_{r})} ; \underline{(n_{1},...,n_{s})}  (4.221)
(if r ≤ s) = 0  (4.222)
(if 1 ≤ s ≤ r) = ∑_{j} (1 − n_{1})^{m_{j(1)}}(1 − n_{2})^{m_{j(2)}} … (1 − n_{s})^{m_{j(s)}} (4.223)

with a sum ranging over all r!/(r − s)! injections j of the set {1, …, s} into the set {1, …, r} and with n^{m} defined in the usual way for vanishing arguments.^{12}

Proofs : derivation of the power constraints.

12ie 0^{0} := 1 ; n^{0} := 1 (∀n ≥ 1) ; 0^{n} := 0 (∀n ≥ 1)

Let t := (t_{1}, …, t_{r}) be s complex variables and define linear maps :

Pow_{[t_{1},...,t_{s}]} : ℍ ⟶ ℂ[t_{1}, …, t_{s}]  (4.224)

by the orthogonality conditions :

< Pow_{[∅)} , ψ\overline{μ} ∂ > = 1  (4.225)
< Pow_{[t_{1}]} , ψ\overline{μ} ∂ > = ∑ t_{1}^{m_{j(1)}}  (4.226)
< Pow_{[t_{1},t_{2}]} , ψ\overline{μ} ∂ > = ∑ t_{1}^{m_{j(1)}} t_{2}^{m_{j(2)}}  (4.227)
…  …
< Pow_{[t_{1},...,t_{s}]} , ψ\overline{μ} ∂ > = ∑ t_{1}^{m_{j(1)}} t_{2}^{m_{j(2)}} … t_{s}^{m_{j(s)}}  (4.228)

for all non-negative partitions \overline{μ} = (m_{1}, …, m_{s}) and with injections j as above. Then the polynomials defined by : P^{n_{1},...,n_{r}}(t_{1}, …, t_{s}) :=< Pow_{[t_{1},...,t_{s}]} , [ψ^{(m_{r})}∂ … [ψ^{(m_{3})}∂, [ψ^{(m_{2})}∂, ψ^{(m_{1})}∂]..] > (4.229)

clearly verify the following induction :

P^{n_{1},...,n_{r}}(∅) := 0 if r ≥ 2  (4.230)
P^{n_{1},...,n_{r}}(t_{1}) := (t_{1} + r − 3) P^{n_{1},...,n_{r−1}}(t_{1}) − P^{n_{1},...,n_{r}}(∅) t_{1}^{n_{r}} (4.231)
…  …
P^{n_{1},...,n_{r}}(t_{1}, …, t_{s}) := (∥t∥ + r − s − 2) P^{n_{1},...,n_{r−1}}(t_{1}, …, t_{s}) +  (4.232)

∑_{1≤k≤s}(∥t∥ + r − s − 2 t_{k}) P^{n_{1},...,n_{r−1}}(t_{1}, …, t̂_{k}, …, t_{s})

with ∥t∥ := ∑ t_{i} and with ˆ in t̂_{k} signalling omission.

The first two relations yield : P^{n_{1},...,n_{r}}(t_{1}) = (t_{1}^{n_{1}} − t_{2}^{n_{2}}) (t_{1} − 1) t_{1} (t_{1} + 1) (t_{1} + 2) … (t_{1} + r − 3) Note the absence of t_{1}^{n_{3}}, t_{1}^{n_{4}} . . .. Therefore :

P^{n_{1},...,n_{r}}(t_{1}) = 0 for r ≥ 2 and r ∈ {1, 0, −1, −2, …, 3 − r} More generally, in view of the factors (∥t∥ + r − s − 2) and (∥t∥ + r − s − 2 t_{k}) in (4.232), it is an easy matter to check, by a double induction on r and s, that the identities :

P^{n_{1},...,n_{r}}(t_{1}, …, t_{s}) = 0

hold true whenever :

r ≥ s + 2 ; t_{k} ∈ {1, 0, −1, −2, −3, … } ; ∑(1 − t_{k}) ≤ r − 2

which establishes the analytic expression of the ‘power constraints’.

4.6 Proofs.

Derivation of the Lagrangian constraints :

The model we used for :

Act := Lie[a, b]/Uker_{∞}  (4.233)

was obtained by specialising (a, b) as (∂_{y} , ψ(y) ∂_{y}). With equal right, we might have specialised (a, b) as (φ(x) ∂_{x} , ∂_{x}). If we now consider the change of variable which takes us from the first to the second specialisation, we find :

x ↦ y = h(x) ; (∂_{y} , ψ(y) ∂_{y}) ↦ (φ(x) ∂_{x} , ∂_{x}) (4.234)
φ(x) = 1/h′(x) ; ψ ∘ h(x)/h′(x) = 1  (4.235)

which is readily seen to imply, for any positive m:

ψ^{(m)} ∂ ↦ \frac{φ^{(m)} ∘ h}{h′} ∂  (4.236)
= (−1)^{m} (φ′)^{m} + (−1)^{m} ∑ lag\underline{ν}(m) φ^{∥\underline{ν}∥} (φ′)^{m−s−∥\underline{ν}∥}φ^{(1+\underline{ν})}

with the same coefficients lag as in (4.214). More generally, we have :

ψ^{(\overline{μ})} ∂ ↦ (−1)^{∥\overline{μ}∥} (φ′)^{∥\overline{μ}∥}  (4.237)
+ (−1)^{∥\overline{μ}∥} ∑ Lag\overline{μ} ; \underline{ν}(m) φ^{∥\underline{ν}∥+1−r} (φ′)^{∥\overline{μ}∥−∥\underline{ν}∥−s}φ^{(1+\underline{ν})}

with the usual notations :

\overline{μ} := \overline{(m_{1}, …, m_{r})} ; ψ^{(\overline{μ})} := ψ^{(m_{1})} … ψ^{(m_{r})}  (4.238)
\underline{ν} := \underline{(n_{1}, …, n_{s})} ; ψ^{(1+\underline{ν})} := ψ^{(1+n_{1})} … ψ^{(1+n_{s})} (4.239)

and with the same tensor Lag\overline{μ} ; \underline{ν}(m) as in (4.211-212). Thus, for the generators ψ^{(\overline{μ})} ∂ of ℍ, the change of variable x ↦ y = h(x) introduces exactly r − 1 negative powers of φ. However, in a single concatenation of Lie brackets :

[ψ^{(m_{r})}∂ … [ψ^{(m_{3})}∂, [ψ^{(m_{2})}∂, ψ^{(m_{1})}∂]..]  (4.240)

no negative powers of φ can possibly appear, since under the change x ↦ y = h(x) the elements inside the multibracket (4.240) transform according to the simple rule :

ψ^{(m_{i})} ∂ ↦ [φ ∂ … [φ ∂, [φ ∂, ∂]..]  (n occurences of φ ∂) (4.241)

There is thus a discrepancy between the behaviour of the generators of ℍ and that of multibrackets, and if we write down the condition for a linear combination of generators to behave like a multibracket, we arrive precisely at the ‘Lagrangian constraints’ of Proposition 4.5 and 4.6.

Equivalence and exhaustiveness of the Lagrangian and power constraints. Dimension of the homogeneous components of Act.

The power constraints are obviously much simpler and far easier to handle than the Lagrangian constraints. Both sets of constraints, however, are equivalent, and with the latter set, exhaustiveness is easier to prove.

The shortest way to proceed is to check, separately, the independence of the various power constraints and that of the various Lagrangian constraints. Then we provisionally assume that the power constraints are exhaustive and see what this says about the dimensions D^{d_{1},d_{2}} of Act. It clearly implies that D^{d_{1},d_{2}} = D_{∗}^{d_{1},d_{2}} − D_{∗∗}^{d_{1},d_{2}}, with D_{∗}^{d_{1},d_{2}} being the dimension of ℍ_{d_{1}}^{d_{1}+d_{2}−1} and D_{∗∗}^{d_{1},d_{2}} the number of (independent !) Lagrangian constraints Lag\underline{ν} subject to (4.214). Therefore :

D_{∗}^{d_{1},d_{2}} = p (d_{1} + d_{2} − 1, d_{1}) = P (2 d_{1} + d_{2} − 1, d_{1})  (4.242)
D_{∗∗}^{d_{1},d_{2}} = 1 + ∑_{{n≤d_{1}−2 \atop n+s≤d_{1}+d_{2}−1}}P (n, s) = p(d_{1} + d_{2} − 1) − p(d_{1} + d_{2} − 1, d_{2})(4.243)

We thus get for D^{d_{1},d_{2}} the equivalent expression (4.200),(4.201) along with the special cases (4.202),(4.203), but under the assumption that the Lagrangian constraints are exhaustive. To fill this one last gape, we introduce on ℍ the filtration ℍ = ∪ ℍ_{p}, where ℍ_{p} denotes the subspace of ℍ generated by products of p Lie elements, and we check that, if the exhaustiveness hypothesis is valid for all components such that d_{1}+d_{2} ≤ d−1, then the next component d_{1} + d_{2} = d has codimension :

codim ℍ_{p}^{d_{1},d_{2}} = ∑_{{n≤d_{1}−2 \atop n+s≤d_{1}+d_{2}−p}} P (n, s) ∀p ≥ 1  (4.244)

But this leaves no scope for ℍ^{d_{1},d_{2}}(= Act^{d_{1},d_{2}}) to have a dimension smaller than D^{d_{1},d_{2}}, which by induction establishes the validity of the expression for D^{d_{1},d_{2}}. But since the power constraints are also mutually independent and, if exhaustive, also lead to the same expression for D^{d_{1},d_{2}}, it means that they, too, must be exhaustive. This completes the proof.

5 Generic, low-complexity identity-tangent twins.

5.1 Finite co-dimension. The general picture.

The basic tools for investigating twin-begetting relations W(A, B) = 1 is not the word W(A, B) itself^{13} but its image w(a, b) in the natural closure of Lie(a, b), defined in the usual way :

w(a, b) := log W(e^{a} e^{b}) = ∑_{m≥1,n≥1} w_{m,n}(a, b) ∈ \overline{Lie}(a, b)  (5.245)

We also require precise information about the nature of the homogeneous components w_{m,n}(a, b) and their exact position with respect to the Ukerfiltration of Lie(a, b). So it is natural to set :

Act_{W} := {(m, n) ∈ ℕ^{2} | w_{m,n}(a, b) ∈ Uker_{0} \ Uker_{∞}}
Act_{W}^{i} := {(m, n) ∈ ℕ^{2} | w_{m,n}(a, b) ∈ Uker_{i} \ Uker_{i+1}} (0 ≤ i)
Act_{W} = Act_{W}^{0} ∪ Act_{W}^{1} ∪ Act_{W}^{2} ∪ …  (5.246)

13at any rate, if we are interested in power series solutions only. But when searching for general transserial twins (see §8), the word W(A, B) returns to the forefront.

Let \overline{Act}_{W} , \overline{Act}_{W}^{0} , \overline{Act}_{W}^{1} … denote the convex hulls of these sets and let Âct_{W} , Âct_{W}^{0} , Âct_{W}^{1} … be the corresponding (finite) sets of extremal points or ‘summits’.

Our twenty examples of §3 were “typical” only in the sense of illustrating the main types of twins that are liable to occurs, with discrete or continuous parameters, built-in symmetries or invariance properties etc. In each case, however, the words W(A, B) consisted of strict multi-commutators, with the result that the set Act_{W} was always included in an upper-right quadrant of summit (d_{1}, d_{2}) ≥ (2, 2) with d_{1}, d_{2} often quite large. As a consequence, w(a, b) had infinitely many vanishing components w_{m,n}(a, b) and the ‘codimension’, in the sense of §2.2, was always infinite.^{14}

Put in simple terms : our twenty ‘typical’ examples were fairly simple, but also embarrassingly non-generic. So we must now tackle the generic case.

In all three tables further down this section each full dot • stands for one homogeneous component Lie^{m,n} and the numbers α^{β^{γ}} next to it indicate the dimension of, respectively, Act_{0}^{m,n}, Act_{∗}^{m,n}, Pass^{m,n}. When γ, or both β and γ, vanish, they simply get omitted.^{15}

5.2 Fixed ratio p/q and no continuous parameter.

Let us first examine the generic counterpart of Examples 1 and 2 in §3.4.

Suppose there are in Âct_{W} two consecutive summits (m_{1}, n_{1}), (m_{2}, n_{2}) which are also in Âct_{W}^{0}. Suppose further that neither of the homogeneous polynomials^{16} w_{m_{1},n_{1}}(p, q) and w_{m_{2},n_{2}}(p, q) has a positive rational root p/q other than p/q = 1. This is automatically the case if (m_{1}, n_{1}) and (m_{2}, n_{2}) are of the form (1, n) or (m, 1) or (2, 2), for the corresponding components of Act^{0} are one-dimensional, but it can occur with any (m, n), for instance for (m, n) = (3, 2) and

14Every twin-begetting word necessarily resolves into a product of simple commutators. But in the generic case, the ‘alternance’ (d_{1}, d_{2}) is (1, 1), which imposes only two constraints: w_{1,0}(a, b) = w_{0,1}(a, b) = 0 (vanishing of the linear components).

15Here, the dimension β of Act_{∗}^{m,n} matters less than the dimension β_{1} of its first subalgebra Act_{1}^{m,n}, but for all points • envisaged, both actually coincide because 0 = β_{2} = β_{3} . . ..

16defined as usual by monomial substitution : w_{m,n}(l_{p}, l_{q}) ≡ w_{m,n}(p, q) l_{mp+nq} with l_{n} := x^{n+1} ∂_{x}.

W(A, B) := A^{c_{1}} B A^{c_{2}} B^{−1} A^{c_{3}} B A^{c_{1}} B^{−1} A^{c_{2}} B A^{c_{3}} B^{−1} (5.247)
w_{2,3}(a, b) = \frac{1}{2} c_{1} c_{2} c_{3} [[a, b], [a[a, b]]]  (5.248)
with  c_{1}, c_{2}, c_{3} ∈ ℤ^{∗} , c_{1} + c_{2} + c_{3} = 0  (5.249)

Suppose lastly that the edge linking the two summits has slope −\frac{p_{0}}{q_{0}} := −\frac{n_{1}−n_{2}}{m_{1}−m_{2}} ≠ −1. The simplest instance of this situation is :

W(A, B) := A B^{3} A B^{−3} A B A^{−3} B^{−1}  (5.250)
w(a, b) = [a[a[a, b]]] + 3 [b[b, a]] + ⋯ = logW(e^{a}, e^{b}) (5.251)

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; 1; 2; 3; 4^{1}; 1; 1; 2; 2; 3; 1; 1; 1; 1; 1; 1; 1; 1; \frac{p}{q} = \frac{1}{2}; codim=4]

5.3 Fixed ratio p/q and one continuous parameter.

Let us now examine the generic counterpart of Examples 3 and 4 in §3.4.

Suppose Âct_{W} has a summit (m_{0}, n_{0}) which is also in Âct_{W}^{0}. Suppose further that the homogeneous polynomial w_{m_{0},n_{0}}(p, q) has one or several positive rational roots p/q (other than p/q = 1) such that the line D of slope −p/q drawn trough the summit (m_{0}, n_{0}) lies outside the convex hull \overline{Act}_{W}. This may occur with any (m_{0}, n_{0}) ≥ (2, 2) and ≠ (2, 2). The simplest instance of this situation corresponds to (m_{0}, n_{0}) = (3, 2) and :

w(a, b) = +c_{1} [b[a[a[a, b]]]] − c_{2} [a[a[b[a, b]]]] + … (5.252)
with \frac{c_{1}}{c_{2}} = \frac{2}{3} \frac{p + 2 q}{p + q}  (5.253)

Exercise : find the simplest words W(A, B) corresponding to the simplest values of p/q.

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; 1; 4; 1; 3; 5^{3^{1}}; 1; 3; 4^{2^{1}}; 6^{5^{3}}; 1; 2; 4^{1}; 5^{2^{1}}; 6^{5^{3}}; 1; 2; 3; 4^{1}; 4^{2^{1}}; 5^{3^{1}}; 1; 1; 2; 2; 3; 3; 4; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; \frac{p}{q} =2,; \frac{3}{2} ,; 1,; \frac{1}{2}; A; B; C; A; B; C; A: codim=8; B: codim=11; C: codim=12]

Drawing a line of slope p/q through the summit (3, 2) within one of the angular domains A, B, C and counting the dimensions α of the cells • below that line, one finds ‘codimensions’ equal to, respectively, 8,11,12.

5.4 Free ratio p/q and one continuous parameter.

Let us now examine the generic counterpart of Examples 5 and 6 in §3.4.

Suppose Âct_{W} has a summit (m_{0}, n_{0}) which lies in Âct_{W}^{1} rather than Âct_{W}^{0}. This may occur with any (m_{0}, n_{0}) ≥ (3, 3) and ≠ (3, 3). The polynomial w_{m_{0},n_{0}}(p, q) is then ≡ 0. Let p/q be any rational ≠ 1 such that the line D of slope −p/q drawn through the summit (m_{0}, n_{0}) lies outside the convex hull \overline{Act}_{W}. The simplest instance of this situation corresponds to (m_{0}, n_{0}) = (4, 3) with an extremal component w_{4,3}(a, b) in Uker_{1} \ Uker_{2}. That space, however, has dimension 1, so that our w_{4,3}(a, b) has to be proportional to :

[[b, \overline{a}^{3} b] \overline{a} b] + 6 [[b, \overline{a} b] \overline{a}^{3} b] − 3 [[b, \overline{a}^{2} b] \overline{a}^{2} b] − 3 [[\overline{a} b, \overline{a}^{2} b] \overline{a} b]

or, equivalently, to :

[[[a, \overline{b}^{2} a] a] \overline{b} a] − 6 [[[a, \overline{b} a] a] \overline{b}^{2} a] − 3 [[a, \overline{b} a] [a \overline{b}^{2} a]] + 4 [[[a, \overline{b} a] \overline{b} a] \overline{b} a]

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; 1; 4; 6^{4^{2}}; 7^{12^{11}}; 1; 3; 5^{3^{1}}; 6^{8^{6}}; 8^{15^{19}}; 1; 3; 4^{2^{1}}; 6^{5^{3}}; 7^{9^{9}}; 8^{15^{19}}; 1; 2; 4^{1}; 5^{2^{1}}; 6^{5^{3}}; 6^{8^{6}}; 7^{12^{11}}; 1; 2; 3; 4^{1}; 4^{2^{1}}; 5^{3^{1}}; 6^{4^{2}}; 6^{6^{3}}; 7^{7^{4}}; 8^{9^{5}}; 1; 1; 2; 2; 3; 3; 4; 4; 5; 5; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; \frac{p}{q} =1; \frac{p}{q} = \frac{2}{3} ,; \frac{p}{q} = \frac{1}{2} ,; A; B; A; B; A: codim=31; B: codim=31]

Drawing a line of slope p/q through the summit (4, 3) within either of the angular domains A, B and counting the dimensions α of the cells • below that line, then adding 4 , which is the dimension of Act_{0}^{4,3} , one finds the same ‘codimension’ with A and B, namely 31.

Exercise : find the simplest words W(A, B) that induce an extremal w_{4,3}(a, b) as above and for which all components lying below the bissectrix of A (or B) vanish.

Remark 1 : additional symmetries. The above constructions in §5.2, §5.3, §5.4 provide generic analogues to the typical situations exemplified in §3, in Ex 1-2, 3-4, 5-6 respectively. As for the other situations reviewed in §3 (Ex 7 through 20) and which mostly involve twins with extra symmetries or invariance properties, they too have their generic counterparts. More precisely, the reader may satisfy himself by going seriatim through these 16 examples that each achievable set of symmetries can also be achieved in finite ‘codimension’.

Remark 2 : precautions with the divisors. The constructions of this section, as indeed the earlier constructions of §3, do work provided the corresponding divisor D(t) doesn’t vanish on ℕ. Simple calculations confirm that this is indeed the case with all three lowestcomplexity examples produced in §5.2, §5.3, §5.4. There exist, however, exceptional cases in which the polynomial D(t) may have k roots on ℕ. Even in such cases, there always exist transserial twins^{17}, but for there to be true power series twins, k additional conditions have to be fulfilled, bearing on the relevant homogeneous components of w(a, b)^{18}. When these conditions are fulfilled, the twins automatically inherit k new continuous parameters.

6 Non identity-tangent twins.

6.1 Twins of tangency (0, q).

Let us now deal with words W(A, B) that are no longer products of commutators (multiple or even simple). As a result, their Lie image w(a, b) is supported by a quadrant of apex (m_{0}, n_{0}) which is no longer ≥ (1, 1) but either (0, 1) or (1, 0). Let us settle for the first choice :

W(A, B) := A^{n_{1}} B^{m_{1}} … A^{n_{r}} B^{m_{r}}  with  (6.254)
n_{j} ∈ ℤ^{∗} , m_{j} ∈ ℤ^{∗} , ∑ n_{j} = 0 but ∑ m_{j} ≠ 0 (6.255)

F^{n_{1}^{⋆}} G^{m_{1}} F^{−n_{1}^{⋆}} … F^{n_{r}^{⋆}} G^{m_{r}} F^{−n_{r}^{⋆}} = 1 with n_{j}^{∗} − n_{j−1}^{∗} = n_{j} (6.256)

with relative integers n_{j}^{∗} defined upto some arbitrary additive constant n_{0}^{∗}. We adjust n_{0}^{∗} so as to have inf(n_{j}^{∗}) = 0 and form the polynomial :

Δ_{W} (θ) := ∑_{j} m_{j} θ^{n_{j}^{∗}}  (Δ_{W}(0) ≠ 0 , Δ_{W} (1) ≠ 0)  (6.257)

These new divisors Δ_{W} (θ) , though rather different in form from the familiar D_{w}(t), play much the same role, since the inductive calculation of the n-th coefficient b_{n} of g leads to relations of the form Δ_{W} (a^{n}) b_{n} = earlier terms. As a consequence, everything hinges on the “resonance” of Δ_{W} (θ), which is the largest integer s for which Δ_{W} (θ) admits s roots of the form θ_{i} = a^{i} , i = 1..s, with s distinct integers q_{i} and a not a unit root. To normalise the situation and fix a, we may, and always shall, assume that the {q_{i}} are co-prime.

17see §8. Here, the relevant transseries closely resemble power series. In fact, they are usually power series of x and some block x (log x)^{s}.

18ie on the w_{m,n}(a, b) for which t = (m − m_{0}) p + (n − n_{0}) q. Of course, that keeps us in “finite codimension”.

Then, depending on whether the resonance k is 1,2,3 et …, our equation (6.256), which may be rewritten as :

G_{n_{1}^{∗}}^{m_{1}} … G_{n_{r}^{⋆}}^{m_{r}} = 1  with  G_{n}^{m} := F^{n} G^{m} F^{−n}  (6.258)

admits a general 1,2,3 … -parameter twin solution of the form :

G = G^{<b_{1}>} := H_{q_{1}}^{b_{1}}  (b_{1} ∈ ℂ , q_{1} ∈ ℕ) (6.259)
G = G^{<b_{1},b_{2}>} := H_{q_{1}}^{b_{1}} H_{q_{2}}^{b_{2}} K^{<b_{1},b_{2}>}  (b_{i} ∈ ℂ , q_{i} ∈ ℕ) (6.260)
G = G^{<b_{1},b_{2},b_{3}>} := H_{q_{1}}^{b_{1}} H_{q_{2}}^{b_{2}} H_{q_{3}}^{b_{3}} K^{<b_{1},b_{2},b_{3}>} (b_{i} ∈ ℂ , q_{i} ∈ ℕ) (6.261)
etc …

with operators H_{q}^{b} and K corresponding to post-composition by mappings of the form :

h_{q_{i}}^{∘b_{i}} : x ↦ x (1 + b_{i} x^{q_{i}})^{−1/q_{i}}  (b_{i} ∈ ℂ , q_{i} ∈ ℕ) (6.262)
k^{<b_{1},b_{2}>} : x ↦ x (1 + O(x^{q_{1}+q_{2}}))  (6.263)
k^{<b_{1},b_{2},b_{3}>} : x ↦ x (1 + O(x^{q_{1}+q_{2}}))  if q_{1} < q_{2} < q_{3} (6.264)
etc …

provided (for k ≥ 3) that the q_{i}’s verify no dependence relation of the form :

q_{i} = ∑_{j≠i} m_{j} q_{j}  with m_{j} ∈ ℕ  and ∑_{m_{j}>0} 1 ≥ 2  (6.265)

For k = 1 the solution (6.259) is of the type which we have dismissed as ‘elementary’ in §1.1. For k ≥ 2, however, the solutions (6.260),(6.261) etc are genuine twins as soon as two of the free complex parameters b_{i} are chosen ≠ 0. Proof: Straightforward : apply Campbell-Hausdorff to rephrase equation (6.258) in terms of the infinitesimal generators G_{n∗}^{m} of the factors G_{n}^{m} and observe that, whenever all parameters b_{i} but one vanish, the ‘corrective factor’ H^{<b_{1},b_{2}>}, H^{<b_{1},b_{2},b_{3}>} … reduces to the identity operator.

6.2 Simple examples.

The simplest possible ‘resonant’ divisor is :

Δ_{W}(θ) := (θ − a^{q_{1}})(θ − a^{q_{2}}) = m_{0} + m_{1} θ + m_{2} θ^{2}  (6.266)
a ∈ ℕ^{∗} , 1 ≤ q_{1} < q_{2} , m_{0} = a^{q_{1}+q_{2}} , m_{1} = −a^{q_{1}} − a^{q_{2}} , m_{2} = 1 (6.267)
e.g. a = 2 , q_{1} = 1 , q_{2} = 2 , m_{0} = 8 , m_{1} = −6 , m_{2} = 1 (6.268)

and may be obtained from these words :

W_{1}(F, G) := G_{0}^{8} G_{1}^{−6} G_{2}  (with G_{n} := F^{n} G F^{−n}) (6.269)
W_{2}(F, G) := G_{1}^{−6} G_{0}^{8} G_{2}  (6.270)
W_{3}(F, G) := G_{0}^{4} G_{1}^{−6} G_{0}^{4} G_{2}  (6.271)
W_{4}(F, G) := G_{1}^{−3} G_{0}^{8} G_{1}^{−3} G_{2}  (6.272)

The last two words have the advantadge of forcing a symmetry within the general twin solution :

{W_{i}(F, G) = 1} ⟺ {W_{i}(F, G^{−1}) = 1} (i = 3, 4)(6.273)
(F, G^{<b_{1},b_{2}>}) twin solution ⟺ (F, G^{<−b_{1},−b_{2}>}) twin solution(6.274)

and of halving the quantity of unknown coefficients to compute.

For the sequel we also require examples with a on the unit circle but not a unit root. The simplest cluster of examples corresponds to divisors of the form :

Δ_{W}(θ) := Δ_{1}(θ) Δ_{2}(θ) = ∑_{0≤i≤4} m_{i} θ^{i} with  (6.275)
Δ_{1}(θ) := l_{0} (θ − a)(θ − ā) = l_{0} + l_{1} θ + l_{0} θ^{2}  (6.276)
Δ_{2}(θ) := l_{0}^{2} (θ − a^{2})(θ − ā^{2}) = l_{0}^{2} + (2 l_{0}^{2} − l_{1}^{2}) θ + l_{0}^{2} θ^{2} (6.277)
for  l_{0} ∈ ℤ^{∗} , l_{1} ∈ ℤ^{∗} , 1 ≠ |l_{1}/l_{0}| < 2  (6.278)
and so  |a| = 1 , a^{n} ≠ 1 (∀n)  (6.279)
m_{0} = m_{4} = l_{0}^{3}  (6.280)
m_{1} = m_{3} = 2 l_{0}^{3} + l_{0}^{2} l_{1} − l_{0} l_{1}^{2}  (6.281)
m_{2} = 2 l_{0}^{3} + 2 l_{0}^{2} l_{1} − l_{1}^{3}  (6.282)

Here are some of the simplest choices :

\begin{matrix}l_{0} = 2 & l_{1} = −1 & m_{0} = 8 & m_{1} = 10 & m_{2} = 9 & m_{3} = 10 & m_{4} = 8 \\ l_{0} = 2 & l_{1} = 1 & m_{0} = 8 & m_{1} = 18 & m_{2} = 23 & m_{3} = 18 & m_{4} = 8 \\ l_{0} = 2 & l_{1} = 3 & m_{0} = 8 & m_{1} = 10 & m_{2} = 13 & m_{3} = 10 & m_{4} = 8 \\ l_{0} = 3 & l_{1} = −2 & m_{0} = 27 & m_{1} = 24 & m_{2} = 26 & m_{3} = 24 & m_{4} = 27 \end{matrix}

The first choices may be realised by either of the words :

W_{5}(F, G) := G_{0}^{8} G_{1}^{10} G_{2}^{9} G_{3}^{10} G_{4}^{8} with G_{n} := F^{n} G F^{−n}(6.283)
W_{6}(F, G) := G_{0}^{4} G_{4}^{4} G_{1}^{5} G_{3}^{5} G_{2}^{9} G_{3}^{5} G_{1}^{5} G_{4}^{4} G_{0}^{4}  (6.284)

Here again, W_{6}, unlike W_{5}, forces a reflection-symmetry of type (6.274).

From the formal viewpoint, the value of a doesn’t matter, as long a is not a unit root, but from the viewpoint of analysis, as we shall see in a moment, the picture changes completely depending on whether a lies on, or outside, the unit circle.

6.3 Twins of tangency (0, 0).

Pick any word W(A, B) leading to twins of tangency order (0, q). Then the word :

W_{∗}(A, B) := W(A, {A, B})  (6.285)

admits a general twin solution of tangency order (0, 0).

To see this, normalise F by setting f(x) := a x for the right value of a.^{19} Next, solve W (F, G_{0}) = 1 with respect to G_{0} as above. Then end by solving G_{0} = {F, G} with respect to G, which is always (uniquely) possible for any choice of b := g′(0), provided b is not a unit root.

6.4 Simple examples.

With the word W_{∗} as in (6.285) and any one of the words W in §6.2.

6.5 Pre-identity-tangent ‘twins’: type (0^{∗}, 0^{∗}).

Let p be prime ≥ 5 and let e_{1}, e_{2} be two distinct unit roots :

e_{1}^{p} = e_{2}^{p} = 1 , e_{1} ≠ 1, e_{2} ≠ 1 , e_{1} e_{2} ≠ 1

Then the system

f := x ↦ e_{1} x + O(x^{2})  (6.286)
g := x ↦ e_{2} x + O(x^{2})  (6.287)
F^{p} = G^{p} = (F G)^{p} = 1  (6.288)

19ie for a ‘resonant’ root of the divisor Δ_{W}(θ).

is easily seen to possess a general non-elementary^{20} solution that inherently^{21} depends on infinitely many parameters. Similar results hold for non-prime powers p ≥ 8, also for a great many systems analoguous to (6.288). But here the dichotomy holds throughout : either there exist no non-elementary solutions, or their exist an infinity of them, with an infinite degree of freedom. This is a sharp departure from the familiar situation, when twins depend only on finitely many parameters. Actually, since (6.288) consists of three equations with just two unknowns, the corresponding pairs (F, G) should not be regarded as proper ‘twins’. As we shall see (§7 infra), their analytic properties also set them apart from genuine twins.

6.6 Simple examples.

With p prime : p = 5 , e_{1} = exp(2πi/5) , e_{2} = exp(4πi/5) With p non-prime : p = 8 , e_{1} = i , e_{2} = −1

6.7 Siblings.

Let us revert to the case of identity-tangent mappings. Proper twins, as we saw, are non-elementary pairs of mappings bound by one relation. The natural generalisation is that of ‘siblings’, or non-elementary ^{22} systems of r mappings bound by r − 1 relations.

1 = W_{1}(F) = … = W_{r−1}(F)  (6.289)
with F := (F_{1}, …, F_{r})

A special but important sub-case arises when we impose the pair-wise commutation of r words V_{i}(F). Indeed, for identity-tangent mappings (or again, for one-variable, formally real mappings), this requirement amounts to just r −1 independent relations rather than r(r −1)/2 , as would be the case with higher-dimensional mappings : see §6.8 infra.

20in the usual sense of not being reducible, even under a joint ramified change of coordinates, to a pair of homographies.

21ie after normalising F, G under a joint change of co-ordinates.

22in the customary sense of not being conjugate to a system of homographies – with allowance, as usual, for ramified changes of co-ordinates.

Let Lie(a) = Lie(a_{1}, …, a_{r}) be the free Lie algebra generated by the a_{i}’s. To each multi-integer p := (p_{1}, …, p_{r}) we associate the projectors p^{∗} and p^{i} defined by specialising as usual the free generators a_{i}’s to ordinary differential operators l_{t} := x^{1+t}∂_{x}:

p^{∗} := Lie(a) → ℤ , w ↦ p^{∗} . w  (6.290)
p^{i} := Lie(a) → ℤ[t] , w ↦ p^{i} . w (1 ≤ i ≤ r) (6.291)

with

p^{∗} . w := \frac{w(l_{p_{1}}, …, l_{p_{r}}).x}{x^{1+p_{1}+···+p_{r}}}  (6.292)
p^{i} . w := \frac{∂_{ϵ} w(l_{p_{1}}, …, l_{p_{i}} + ϵ l_{t+p_{i}} …, l_{p_{r}}).x ||_{ϵ=0}}{x^{1+t+p_{1}+···+p_{r}}}  (6.293)

If w ∈ Lie(a) is homogeneous of degree d_{1} in a_{1}, d_{2} in a_{2}, …, then its p-degree is simply defined as p-deg(w) := ∑ p_{i} d_{i}.

Clearly, if w is of p-degree P, these two identities hold :

(t − P) p^{∗}.w = ∑_{1}^{r} (t − p_{i}) p^{i}.w  (6.294)
p^{i}.w = d_{i} p^{∗}.w + O(t) as t → 0  (6.295)

Now, fix a multi-integer p := (p_{1}, …, p_{r}) and a shorter (one element less!) sequence w := (w_{1}, …, w_{r−1}) of homogeneous elements of Lie(a), of various or identical p-degrees, it doesn’st matter, but all subject to the orthogonality condition :

p^{∗} . w_{1} = p^{∗} . w_{2} = ⋯ = p^{∗} . w_{r−1} = 0  (6.296)

In view of (6.294), (6.295) the relation :

D(t) := \frac{(−1)^{i}}{(t − p_{i})} det[p^{1}.w, …, { \atop p^{i}.w}, …, p^{r}.w]  (6.297)

defines a function D(t) = D_{w}(t, p) that is not only independent of i ^{23} but also polynomial in t. This function is the exact generalisation for siblings of the ‘divisors’ which we came up against when discussing twins.

23the term { \atop p^{i}.w} is omitted in (6.297) and of course p^{j}.w stands for [p^{j}.w_{1}, …, p^{j}.w_{r−1}]

Proposition 6.1 Siblings. Consider a system of type (6.289) and fix a multi-integer p. Assume that each Lie element w_{i} has exactly one lowest-degree homogeneous component \underline{w}_{i}. In other words :

w_{i}(a) := log W_{i}(e^{a}) = \underline{w}_{i}(a) + terms of higher p-degree (i = 1 … r) (6.298)

Then, for the system (6.289) to admit a non-elementary ‘sibling’ solution of the form

f_{i}(x) = x (1 + a_{i} x^{p_{i}} + ∑_{t≥1} a_{i}(t) x^{t+p_{i}}) (i = 1 … r)  (6.299)

with t in ℕ or, what here amounts to the same, in p.ℕ := p_{1} ℕ+⋯+p_{r} ℕ, it is necessary that the orthogonality conditions p . \underline{w}_{i} = 0 be fulfilled for each i and, when they are, it is sufficient that the corresponding divisor D(t) = D\underline{w}(t, p) should be ≢ 0 for t ∈ p.ℕ.

We skip the proof, as it hardly differs from the one for twins (r = 2).

6.8 Simple examples.

Since we haven’t developped the Uker-graduation of algebras Lie(a_{1}, …, a_{r}) for r ≥ 3, we shall have to be content with a typical rather than a generic example, ie one relying on highly multiple commutators and, consequently, of infinite ‘codimension’. Moreover, to bring home the point that, with siblings as with twins, one can achieve invariance under any given periodic automorphism^{24} of the r-generator free group, we shall choose r = 3 and pick the simplest example with free orders of tangency (p_{1}, p_{2}, p_{3}) and invariance under the circular permutation (F_{1}, F_{2}, F_{3}) ↦ (F_{2}, F_{3}, F_{1}). First we set :

V(A_{1}, A_{2}, A_{3}) := {U(A_{3}, A_{1}), U(A_{2}, A_{1})}  (6.300)
with U(B, A) := A{B^{−1}{B, A^{2}}}A^{−1}  (6.301)
= A B A^{−2} B^{−1} A^{2} B^{−1} A^{−2} B A  (6.302)
= (U(B^{−1}, A)^{−1})^{−1}  (6.303)

Now, in a general free group setting, imposing the pair-wise commutation of the three words :

V(A_{1}, A_{2}, A_{3}) , V(A_{2}, A_{3}, A_{1}) , V(A_{3}, A_{1}, A_{2})  (6.304)

24or any given group of periodic automorphisms

would amounts to imposing three independent relations. However, when dealing with identity-tangent mappings F_{i}, imposing the pair-wise commutation of the three words :

V_{1} := V (F_{1}, F_{2}, F_{3}) , V_{2} := V (F_{2}, F_{3}, F_{1}) , V_{3} := V (F_{3}, F_{1}, F_{2}) (6.305)

amounts to imposing only two independent relations, e.g. :

W_{12}(F_{1}, F_{2}, F_{3}) := {V_{1}, V_{2}} = 1  (6.306)
W_{23}(F_{1}, F_{2}, F_{3}) := {V_{2}, V_{3}} = 1  (6.307)

This ‘transitiveness’ property of commutation for identity-tangent mappings is simply due to the fact that F, G do commute if and only if their logarithms F_{∗} := log(F) = f_{∗}∂_{x} , G_{∗} = log(G) = g_{∗}∂_{x} coincide, as operators, upto multiplication by a constant.

Thus we have our system of 2 relations (6.326),(6.307) with three unknowns F_{i}, and we can easily see than it admits a general sibling solution with free tangency orders (p_{1}, p_{2}, p_{3}) by observing that :

log V (F_{1}, F_{2}, F_{3}) = −2 p_{1} p_{1} (p_{1}−p_{2}) (p_{2}−p_{3}) (p_{3}−p_{1}) x^{1+2p_{1}+2p_{2}+2p_{3}}∂_{x}

+ terms of higher degree

and then reasoning as in §3.4, Example 5 or 6.

7 Analytic nature of twins.

7.1 Multiplier |a| ≠ 1: convergence.

Let us start with the case which, from the analytic viewpoint, is simplest : that of a multiplier |a| ≠ 1. We first require an auxiliary lemma.

The ‘sandwich equation’: Consider the equation :

H^{m_{1}} K_{1} H^{m_{2}} K_{2} … H^{m_{r−1}} K_{r−1} H^{m_{r}} K_{r} = 1  (7.308)
h(x) unknown ∼ a x  (7.309)
k_{i}(x) given ∼ b_{i} x and analytic  (7.310)
m_{1} + ⋯ + m_{r} ≠ 0 m_{i} ∈ ℤ  (7.311)
a^{m_{1}+···+m_{r}} b_{1} … b_{r} = 1  (7.312)

which – conditions (7.309-312) aside – is the most general type of equation that can arise in a non-abelian group. It goes by the name of ‘sandwhich equation’ (see [F]) because the iterates of the unknown mapping h alternate with given factors k_{i}. However, the assumption ∑ m_{i} ≠ 0 is really one of non-alternance. It says that the right-hand (7.308) doesn’t resolve into a product of commutators, so that the sandwich equation is truly the ‘opposite’ of the highly alternate equations which we have been reviewing in our search for identity-tangent twins. It is, however, quite relevant for the non identity-tangent case, in particular for proving analyticity.

Changing both data and unknown H ↦ \underline{H} H , K_{i} ↦ \underline{K}_{i} K_{i}, with suitabble analytic \underline{H}, \underline{K}_{i} we get an equivalent equation :

H_{(0)}^{ϵ_{0}} B_{0} H_{(1)}^{ϵ_{1}} B_{1} … H_{(l−1)}^{ϵ_{l−1}} B_{l−1} H_{(l)}^{ϵ_{l}} B_{l} = 1 with  (7.313)
H_{(i)} := C_{i} H C_{i}^{−1}  (7.314)
h(x) new unknown ∼ x  (7.315)
b_{i}(x) new analytic data ∼ x  (7.316)
c_{i}(x) new analytic data ∼ γ_{i} x  (7.317)
ϵ_{1} + ⋯ + ϵ_{l} ≠ 0 ϵ_{i} ∈ {1, −1}  (7.318)

Upto reindexation and multiplication by a common constant, the c_{i} depend only on the m_{i} and a_{i}. If |γ_{i}| attains its maximum for exactly one index i (let us call this the “peak condition”), we can show that the sandwich equation admits a (clearly unique) analytic solution.

Proof : We may always arrange for the peak value |γ_{i_{0}}| to be 1 and re-index so as to have i_{0} = 0. We may also arrange for C_{0} = 1. Eventually, our H is the unique solution of the fixed point problem

H = P (H) with P (H) := (B_{0} H_{(1)}^{ϵ_{1}} B_{1} … H_{(l−1)}^{ϵ_{l−1}} B_{l−1} H_{(l)}^{ϵ_{l}} B_{l})^{−1} (7.319)

The number of factors in P^{n} (1) is exactly σ(l, n) := \frac{l+1}{l−1} (l^{n} − 1). Moreover, as soon as n exceeds n_{0}, the factors of the form

B_{i(j_{1},j_{2}...,j_{n_{0}})}^{±1} := C_{j_{n_{0}}} … C_{j_{2}} C_{j_{1}} B_{i}^{±1} C_{j_{1}}^{−1}C_{j_{2}}^{−1} … C_{j_{n_{0}}}^{−1}  (7.320)

present inside P^{n} (1) do not change, although their exact location does. Their total number is clearly (l + 1) l^{n_{0}}. Their total contribution, therefore, will be small of order (l + 1) l^{n_{0}} γ_{∗}^{n_{0} k_{∗}} with γ_{∗} := inf|γ_{i}| < 1 for i = 1..l and with k_{∗} > 1 denoting the lowest tangency order for the new unknown h and the new data b_{i}. But if we choose \underline{H} analytic and close enough (ie tangent to a high enough order) to a formal solution of H_{0}, we can make k as large as we wish and in particular ensure that |l γ_{∗}^{k}| < 1. Fixing such a k_{∗}, we see that the analytic operators P^{n}(1) converge normally to an analytic operator P^{∞}(1) which necessarily coincides with H_{0}. Which means that the formal H_{0} was analytic in the first place. QED.

Application to twins : The scheme clearly applies to all four examples W_{1}(A, B) … W_{4}(A, B) above since m_{2} = 1. It would apply equally well to any example constructed from a monic polynomial^{25} Δ_{W}(θ) with at least one ‘resonant’ root |a| > 1. It applies, in fact, with minor adaptations, to most situations involving multipliers |a| ≠ 1.

An interesting aside is this : once f has been normalised to f(x) = a x, what can be said of g and its natural Riemann surface ? The latter appears to possess, for almost all values of the parameters b_{i}, a highly fractal boundary.

7.2 Multiplier |a| = 1: divergence.

When a is on the unit circle but not a unit root, we should expect generic divergence of the formal twins since, in a suitable z-chart (z ∼ ∞), solving the twin equation reduces to solving an infinite sequence of affine equations :

P φ_{n} = ψ_{n} with P φ(z) := ∑ c_{k} φ(a^{k} z + b_{k}) and z ∼ ∞ (7.321)

where, at the n-th inductive step, ψ_{n} is known and φ_{n} unknown ^{26}. Now, even for an analytic^{27} input ψ_{n}, the solution φ_{n} is generically divergent and non-summable. Of course, compensation within the series ∑ φ_{n} cannot be ruled out off hand, but when the solution depends on a continuous parameter, a simple argument shows that we must have divergence for almost all values of that parameter.

25ie a polynomial with integer coefficients and a leading term with unit coefficient.

26The sum ∑ in the definition of P is finite but it never reduces to a single term.

27at infinity.

7.3 Pre-identity-tangent ‘twins’: convergence.

As already pointed out, these are not genuine twins, since they are constrained by a system S of more than one relation and depend on countably many parameters. However, that very circumstance makes it possible to construct analytic solutions. The proof, which we skip, relies on the fact :

– that S resolves itself into an inductive sequence of affine systems S_{1}, S_{2} …, each of which admits analytic solutions

– that suitable a priori bounds can be imposed on the solution of S_{n} to ensure convergence in the resulting series.

Needless to say, there doesn’t seem to exist any ‘privileged’ choice among these analytic solutions.

7.4 Identity-tangent twins : the general picture.

This is by far the most interesting, but also the most tricky case. The broad picture with identity-tangent twins of tangency order (p, q), ie of the form :

f(x) = x (1 + α x^{p} + …) , g(x) = x (1 + α x^{q} + …) (α ≠ 0, β ≠ 0)

with, say, p < q, is as follows :

P1 : Due to twins (f, g) being defined only upto a joint conjugation, the question that makes sense is : how simple (convergent, Gevrey, resurgent, etc) can f and g be rendered simultaneously ? Another way of approaching the problem is to normalise f to f^{nor} and then look at the nature of the corresponding conormal form g^{conor} of g. Or vice versa. Or again, more intrinsically, one may investigate the connector h^{nor} defined in (3.14). As it happens, it is enough to understand any one of these functions to understand the rest ^{28} and so we shall refer to them collectively as “the intrinsic twin functions”.

P2 : The intrinsic twin functions are doubly resurgent, with respect to the slower critical time z_{1} := x^{−p} and to the faster critical time z_{2} := x^{−q}.

P3 : The intrinsic twin functions are accelero-summable. Moreover, the axis ℝ^{+} is free of singularities in either of the Borel planes {ζ_{1}} and {ζ_{2}} conjugate to the multiplicative plane {z_{1}} and {z_{2}}. As a consequence there exists a privileged real accelero-summation.^{29}

28this is by no means obvious, since the normalisation of an analytic identity tangent f to f^{nor} usually involves a divergent-resurgent change of coordinate h: see §1.3. In the present instance, however, the divergence-resurgence of h cannot ‘add’ to the divergenceresurgence already present in g.

P4 : The privileged sum thus obtained coincides with the one produced by the ‘geometric’ method sketched in §8.4.

P5 : The invariants associated with the z_{1}- and z_{2}-resurgence regroup naturally to form a geometric object, the “shadow twins”, consisting of two formal series^{30} also connected by one relation. These “shadow twins” carry all the obstructions to the analyticity of the original twins (f, g).

P6 : Whenever twins depend on one or several continuous parameters, the intrinsic twin functions are guaranteed to be non-analytic (ie strictly resurgent) except (at most) for a discrete set of parameter values.

P7 : It would seem reasonable to conjecture the non-analyticity of all genuine^{31} identity-tangent twins.

7.5 Identity-tangent twins : autonomous differentialdifference equations and double resurgence.

The phenomenon of double resurgence in the ‘intrinsic twin functions’ is simplest to understand in cases with a free ratio p/q and a free continuous parameter γ = α/β, like in Ex 5 and Ex 6 of §3.4. To further simplify, we impose symmetries that make the iterative residues α_{∗}, β_{∗} of F, G vanish^{32} so that F and G may be separately normalised to :

F^{nor} = exp(−αx^{p+1} ∂_{x}) G^{nor} = exp(−βx^{q+1} ∂_{x}) (γ := α/β) (7.322)

29consisting in calculating on ℝ^{+} both the acceleration integral in the ζ_{1}-plane and the Laplace integral in the ζ_{2}-plane.

30they are not power series, though, and, unlike the original twins, they carry transcendental rather than rational coefficients

31ie non-elementary, in the sense of §1.1

32see (1.19).

Here is one such example :

W(A, B) := {U(A, B), V(A, B)} = 1 with
U := U^{++} U^{+−} U^{−+} U^{−−}
V := V^{++} V^{+−} V^{−+} V^{−−}
U^{++}(A, B) := B {{B{B, A}}, {B, A}} B^{−1}
V^{++}(A, B) := B {B{A{B{B, A}}}} B^{−1}
U^{−+}(A, B) := U^{++}(A^{−1}, B)
U^{+−}(A, B) := U^{++}(A, B^{−1})^{−1}
U^{−−}(A, B) := U^{++}(A^{−1}, B^{−1})^{−1}
V^{−+}(A, B) := V^{++}(A^{−1}, B)
V^{+−}(A, B) := V^{++}(A, B^{−1})^{−1}
V^{−−}(A, B) := V^{++}(A^{−1}, B^{−1})^{−1}

Going over to the Lie algebra, we find :

u(a, b) := log U(e^{b}e^{b}) = 4 [\overline{b} \overline{b}a, \overline{b}a] + higher degree ≥ 7 (7.323)
v(a, b) := log V(e^{b}e^{b}) = 4 [\overline{b} \overline{a} \overline{b} \overline{b}a] + higher degree ≥ 7 (7.324)

However, whereas in the formal investigation it was permissible to linearise with respect to A and B simultaneously, the analytic investigation must take global effects into account. More precisely, in order to understand the double resurgence, with must linearise with respect to A alone, then B alone, and then, in order to establish accelerablity, we must see how these two resurgences interact.

First linearisation : We first linearise in A, i.e. in F. We find in the groups of diffeos :

U^{++}(F, G) = F_{1}^{−1} F_{0} F_{−1}^{−1} F_{0} F_{1}^{−1} F_{−1} F_{0}^{−1} F_{1} F_{0}^{−1} F_{1}
V^{++}(F, G) = F_{0}^{−1}F_{−1}F_{−2}^{−1}F_{−1}F_{0}^{−1}F_{−1}^{−1}F_{−2}F_{−1}^{−1}F_{0}^{2}F_{1}^{−2}F_{0}F_{−1}^{−1}F_{0}F_{1}F_{0}^{−1}F_{−1}F_{0}^{−1}F_{1}
with F_{n} := G^{n} F G^{−n}

and then in the Lie algebra :

u(F, G).x = log(U(F, G)).x = 4 Φ_{−1,0} − 4 Φ_{−1,1} + 4 Φ_{0,1} + deg ≥ 4 (7.325)
v(F, G).x = log(V (F, G)).x = −2 Φ_{−1,0} + Φ_{−1,1} − 2 Φ_{0,1} + deg ≥ 4(7.326)

with and

F_{∗} := log F = Φ(x) ∂_{x}
F_{∗ n} := log F_{n} = Φ_{n}(x) ∂_{x} = \frac{Φ ∘ g^{∘n}(x)}{∂_{x} g^{∘n}(x)} ∂_{x}
Φ_{m,n} := [Φ_{m}, Φ_{n}] := Φ_{m} Φ′_{n} − Φ′_{m} Φ_{n}

Switching over to the ‘critical variable’ z_{2} := x^{−q} that normalises G we get :

G_{∗} := log G = ∂_{z_{2}}  ⇒  g(z_{2}) := z_{2} + 1
F_{∗} := log F = φ(z_{2}) ∂_{z_{2}}
F_{∗ n} := log F_{n} = φ_{n}(z_{2}) ∂_{z_{2}} = φ(z_{2} + n) ∂_{z_{2}}
φ_{m,n} := [φ_{m}, φ_{n}] = φ(z_{2} + m) φ′(z_{2} + n) − φ′(z_{2} + m) φ(z_{2} + n)

Freezing to β = 1 the parameter associated with G and expanding everything in powers of the parameter α associated with F, we find :

φ(z_{2}) = α φ^{<1>}(z_{2}) + α^{2} φ^{<2>}(z_{2}) + … with
P_{2} φ^{<1>} = 0  (7.327)
P_{2} φ^{<k>} = earlier terms (k = 2, 3 …)  (7.328)

with a bilinear difference-differential operator P_{2} of the form :

P_{2} φ(z_{2}) := ∑_{m<n} c_{m,n} [φ(z_{2} + m), φ(z_{2} + n)] and

and coefficients c_{m,n} affine in σ:=p/q and easily calculable from the linearisations (7.325),(7.326) :

c_{−2,0} = c_{0,2} = −\frac{1}{2}(σ − 1)
c_{−1,0} = c_{0,1} = 2 (2σ + 1)
c_{−1,1} =  −3 (σ + 1)

Second linearisation : If we now linearise in B i.e. in G, the picture remains much the same, with (F, Φ, φ, α, z_{2}, σ, P_{2}) and (G, Ψ, ψ, β, z_{1}, σ^{−1}, P_{1}) exchanging places, but with a trilinear operator P_{1} instead of the bilinear P_{2}. Let us, for definiteness, go through this second list of parallel equations. In the group we get :

U^{++}(F, G) = G_{0} G_{−1}^{−1} G_{0}^{−1} G_{−1} G_{0} G_{−1}^{−2} G_{0} G_{−1} G_{0}^{−1} G_{−1} G_{0}^{−1}
V^{++}(F, G) = G_{−1}^{−1} G_{0}^{−1} G_{−1} G_{0} G_{−1}^{−1} G_{−2}^{−1} G_{−1} G_{−2} G_{0}
G_{−2}^{−1} G_{−1}^{−1} G_{−2} G_{−1} G_{−1}^{−1} G_{−1}^{−1} G_{0} G_{−1} G_{0}^{−1}
with G_{n} := F^{n} G F^{−n}

Going over to the Lie algebra, these relations become :

u(F, G).x = log(U(F, G)).x = ∑_{n_{1},n_{2},n_{3}} c_{n_{1},n_{2},n_{3}}^{∗} Ψ_{n_{1},n_{2},n_{3}} + deg ≥ 6(7.329)
v(F, G).x = log(V (F, G)).x = ∑_{n_{1},n_{2},n_{3}} c_{n_{1},n_{2},n_{3}}^{∗∗} Ψ_{n_{1},n_{2},n_{3}} + deg ≥ 6(7.330)

when expressed in the given chart x:

G_{∗} := log G = Ψ(x) ∂_{x}
G_{∗ n} := log G_{n} = Ψ_{n}(x) ∂_{x} = \frac{Ψ ∘ f^{∘n}(x)}{∂_{x} f^{∘n}(x)} ∂_{x}
Ψ_{n_{1},n_{2},n_{3}} := [[Ψ_{n_{1}}, Ψ_{n_{2}}]Ψ_{n_{3}}]

Both sums ∑ in (7.329),(7.330) are finite, and their coefficients c_{n_{1},n_{2},n_{3}}^{∗} , c_{n_{1},n_{2},n_{3}}^{∗∗} are easily calculable from the above factorisations of U^{++}, V^{++}.

Turning to the critical chart z_{1} := x^{−p} that normalises F , we find :

F_{∗} := log F = ∂_{z_{1}}  ⇒  f(z_{1}) := z_{1} + 1
G_{∗} := log G = ψ(z_{1}) ∂_{z_{1}}
G_{∗ n} := log G_{n} = ψ_{n}(z_{1}) ∂_{z_{1}} = ψ(z_{1} + n) ∂_{z_{1}}

the components of the β-expansion of ψ are recursively given by the system :

ψ(z_{1}) = β ψ^{<1>}(z_{1}) + β^{2} ψ^{<2>}(z_{1}) + …
P_{1} ψ^{<1>} = 0  (7.331)
P_{1} ψ^{<k>} = earlier terms  (k = 2, 3 …)  (7.332)

with a difference-differential operator P_{2} of the form (finite sum) :

P_{1} ψ(z_{1}) := ∑_{n_{1},n_{2},n_{3}} c_{n_{1},n_{2},n_{3}} [ψ(z_{1} + n_{1})[ψ(z_{1} + n_{2}), ψ(z_{1} + n_{3})]]

whose coefficients c_{n_{1},n_{2},n_{3}} are affine in σ:=p/q and easily deducible from the linearisations (7.329),(7.330) :

c_{n_{1},n_{2},n_{3}} ≡ 2 (2 σ + 1) c_{n_{1},n_{2},n_{3}}^{∗} + (σ − 1) c_{n_{1},n_{2},n_{3}}^{∗∗}

Singular autonomous difference-differential operators P : The homogeneous equations (7.327) or (7.331) which start the induction can easily be shown to admit a unique formal solution of the form :

φ^{<1>}(z_{2}) = z_{2}^{1−σ} (1 + ∑ γ_{2n}(σ) z_{2}^{−2n}) (σ := p/q)  (7.333)
ψ^{<1>}(z_{1}) = z_{1}^{1− \frac{1}{σ}} (1 + ∑ δ_{2n}(σ) z_{1}^{−2n}) (σ^{−1} := q/p) (7.334)

that is divergent-resurgent in its single “critical variable”, z_{2} or z_{1}, and always possesses a countable infinity of non-zero alien derivatives.

The same holds for the non-homogeneous equations (7.328) or (7.332) that continue the induction, as long as their right-hand sides are themselves resurgent.

Here, the ‘invariants’ or ‘resurgence coefficients’ or ‘Stokes constants’ which enter the resurgence equations as their only transcendental ingredient, are particularly interesting entire functions of σ, of so-called autarkic type^{33}.

7.6 Identity-tangent twins : canonical accelero-summability.

Thus, things are fairly unproblematic as long as we expand our ‘intrinsic twin-related functions’ in powers of the free parameter and consider each contribution φ^{<k>} or ψ^{<k>} in isolation. But the moment we attempt to sum all these contributions, a sharp dissymmetry makes itself felt between the two critical variables. Consider for definiteness the choice p < q ^{34} so that here z_{1} is the ‘slower’ and z_{2} the ‘faster’ variable : z_{1} << z_{2}.

33ie with an asymptotics completely defined by a finite set of – equally autarkic – entire functions. Actually, the class of autonomous difference-differential equations and their invariants would warrant a special investigation, but there is no room for that here.

Summing φ(z_{2}) in α is no problem formally, since only a finite number of terms φ^{<k>}(z_{2}) contribute to any given (negative) power of z_{2}, but Borel transforming φ(z_{2}) to φ̂(ζ_{2}) leads to a divergent power series, which is no use at all.

On the other hand, summing ψ(z_{1}) in β and Borel transforming it to ψ̂(ζ_{1}) yields a convergent germ at ζ = 0, with a sum that can be analytically continued along the whole of ℝ^{+} with a faster-than-exponential growth pattern :

|ψ̂(ζ_{1})| < C_{0} exp( C_{1} |ζ_{1}|^{\frac{q}{q−p}}) (0 < C_{i} < +∞)  (7.335)

This is not good enough to apply the Laplace transform^{35} but just about right to apply the acceleration integral ^{36} from ζ_{1} to ζ_{2} and get a new function that now has exponential growth in ζ_{2} and so can be subjected to Laplace integration.

Accelero-summation here is straightforward, as there are no singularities over the axis arg(ζ_{1}) = 0 or arg(ζ_{2}) = 0 in either of the Borel planes. All told, we get a canonical sum, resulting in ‘germinal’ twins (f, g) that are defined and regular in a sectorial neighbourhood of ℝ^{+} in the original geometric plane x.

For an alternative method – which in this case leads to the same result, and is more geometric, but not nearly as constructive – see §8.4 infra.

7.7 Identity-tangent twins : ‘shadow twins’ and obstructions to analyticity.

There is a standard way of attaching to any resurgent or multi-resurgent function f two richer objects, display(f) and restrict(f), which automatically satisfy the same relations^{37} as f, while “displaying” in user-friendly manner all the resurgence properties of f and all its transcendental ‘invariants’ or ‘Stokes constants’. The display carries all alien derivations of f, of all orders, and combines the original, resurgence-bearing variable z with a countable infinity of so-called pseudo-variables Z^{ω_{1},...,ω_{r}}. The restriction is obtained therefrom by jettisoning the true variable and retaining only the pseudo-variables. For details, see for instance [E9], §2.4.

34remember that here the ratio p/q is free.

35which reverses Borel

36see [E6], Lecture 3, p114-115.

37whatever their form or number, provided they make use “only” of the operations +, ×, ∘, ∂ and their inverses.

Applying this to a pair of twins (f, g) ^{38} linked by a relation (7.336), we get first a mixed object (7.337), and then a ‘pure’ one, the ‘shadow twins’, linked by a relation (7.338) formally identical to (7.336), and which concentrates all the obstructions to the analyticity of (f, g).

twins : 1 = W (f, g)  ⟹  (7.336)
1 = W (display(f), display(g)) ⟹  (7.337)
shadow twins : 1 = W (restrict(f), restrict(g))  (7.338)

8 Transserial twins.

8.1 Reminders about transseries.

This section is something of an aside, and we shall be extremely sketchy.

Let z ∼ +∞. The formal trigebra 𝕋 or ℝ[[[z]]] consists, very roughly, of the natural completion of ℝ[[z^{−1}]] under the basic operations {+, ×, ∘, ∂} and their inverses. Its elements, the so-called transseries, may involve extremely intricate concatenations of exponentials and logarithms, but always admit a (unique) distinguished or ‘canonical’ representation obtained by

– expelling all infinitesimals from inside the exponentials

– expelling all sums from inside the logarithms^{39} There is a natural, total order on the trigebra of transseries. In particular,

in the distinguished representation, each transseries appears in the guise of a transfinite, well-ordered sum of transmonomials which, despite their ‘atomicity’, may carry infinitely many coefficients, with a complex arborescent structure on them.

So much for the formal transseries. The geometric counterpart is the trigebra ℝ{{{z}}} of so-called analysable germs at +∞. These are in one-to-one correspondance with a small (or huge, depending on how you look at it) subalgebra of ℝ[[[z]]], consisting either of ‘convergent’ or divergent but ‘accelero-summabable’ transseries.

38expressed in any chart that makes them simultaneously resurgent.

39using in both cases the functional equations and Taylor expansions of exp and log.

The correspondance, needless to say, commutes with the full array of basic operations {+, ×, ∘, ∂} and respects the natural order. Both trigebras enjoy a long list of auspicious properties : stability, closure, algorithmic solvability of many types of differential or functional equations, etc.

A number of applications, though, call for even larger extenstions, resulting from the introduction of extremely fast-growing functions E_{n}, the so-called transexponentials of strength n, or the even faster growing ultraexponential E_{∞}.

Before defining the latter, let us recall the usual abbreviations for commutators and conjugations :

\overline{\overline{A}}.B := \{A, B\} := A^{−1} B^{−1} A B  (8.339)
A : B := B A B^{−1}  (8.340)
A : B : C := A : (B : C)  (8.341)
≠ (A : B) : C = A : (C B)

Let T be the unit shift and E the exponential mapping. The transexponentials E_{n} are ‘characterised’ by “smoothness” and the conjugation equations :

T : E_{1} ≡ E
T : E_{2} ≡ E_{1}
T : E_{3} ≡ E_{2}
…

The ultraexponential E_{∞} is also ‘characterised’ by “smoothness” and a growth pattern faster than that of any E_{n} of finite strength n.

Actually, no matter what amount of “smoothness” we choose to impose, there is a huge indeterminacy inherent in this construction ^{40}, which cannot be removed by purely asymptotic criteria at +∞. This is the bad news, or the ‘negative side’ of the so-called indiscernibility theorem. However, by the same token, no matter what choice we strike, the resulting trans- or ultra-exponential extensions of our trigebra of transseries are going to be essentially one, ie isomorphic under the full struxture {+, ×, ∘, ∂, <}. This is the good news, or the ‘positive side’ of the indiscernibility theorem.

40roughly, one smooth periodic conjugation per step, and an even more formidable indeterminacy for E_{∞}.

The trigebras of transseries and analysable functions were introduced in the late 90s to solve the so-called Dulac problem about the finiteness of limit-cycles. See [E5], [E6], [E7], [E8]. The emphasis there was squarely on the analytic side, and developments about the formal construction – ie the trigebra of transseries – were kept to a minimum. A far more sophisticated theory of formal transseries was subsequently developed by van der Hoeven (see [H1], [H2], [H3]), with special attention to the algorithmic resolution of differential, functional etc equations. The present section §8, incidentally, benefited from exchanges we had with van der Hoeven. Lastly, for complements about the indiscernibility theorem, the fast/slow functions (such as the trans- or ultra-exponentials and their reciprocals) and the whole subject of ‘universal fast/slow asymptotics’, we refer to [E5], chap. 7-10.

8.2 Transserial twins of exponential or transexponential type.

Switching from power series to transseries, especially of the transexponential sort, brings significant changes to the typology of twins. There appears a whole new class of twin-begetting relations which may be be called ‘regular’ or ‘orderly’ and which had no true equivalent in the more restrictive setting of power series.

Consider for instance the following series of (F, G)-relations, where F has been normalised to the unit shift T. They consist of a principal, highlyalternate, (1 + k)-shrinking^{41} factor {T , T^{n_{1}} : ⋯: T^{n_{k}} : G} and a perturbative, even more alternate, (2 + k)-shrinking factor. W_{k}(T, G).

twin relation  nature of G  par. nb.
1 = W_{0}(T, G){T , G}  translations  1 param.
1 = W_{1}(T, G){T , T^{n_{1}} : G}  dilatations  2 param.
1 = W_{2}(T, G){T , T^{n_{1}} : T^{n_{2}} : G}  exponentials  3 param.
1 = W_{3}(T, G){T , T^{n_{1}} : T^{n_{2}} : T^{n_{3}} : G} transexp. of str. 1  4 param.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 = W_{k}(T, G){T , T^{n_{1}} : ⋯: T^{n_{k}} : G} transexp. of str. k−2 k param.

41an operator G ↦ P(G) is k-shrinking if, ∀m > 0 , it ‘shrinks’ any transexponential of strength m + k to something that lies in the transexponential growth range of strength m.

The k-th equation of this list has a neat, predictable set of twin solutions, which constitute a k-dimensional real variety.

Then, alongside these ‘regular’ or ‘orderly’ twin-begetting relations, we have also, like with power series but in an even more bewildering variety of shapes, a collection of ‘exceptional’ or ‘sporadic’ relations, with twin solutions whose existence, number, nature, connectedness/disconnectedness etc … depend on a ‘fluke of algebra’ – like logW(e^{a}, e^{b}) accidentally falling into this or that cell of the Uker-filtration – that is not at all apparent on the word W(A, B) itself.

8.3 Van der Hoeven’s intermediate value theorem.

Let P be a differential polynomial with transserial coefficients. Van der Hoeven has shown in [H2] that, given any pair of transseries g_{1}, g_{2} such that P (g_{1}) < 0 < P (g_{2}) there exists at least one solution P (g_{0}) = 0 with g_{0} ∈]g_{1}, g_{2}[. He even supplies an algorithm for constructing g_{0}. He has also extended this intermediate value theorem to an even broader class of functional equations – not broad enough, however, to cover all composition equations W (f, g) = id^{42}. Nevertheless, it seems highly likely that the theorem holds in that context also.

8.4 Canonical resummation of transserial twins.

Fundamental squares : Let W (f, g) = id be a twin-begetting relation and (f̃_{0}, g̃_{0}) a formal transserial solution, normalised to f̃_{0} = T = unit shift. We wish to “sum” (f̃_{0}, g̃_{0}) to a geometric twin (f_{0}, g_{0}) consisting of true germs at +∞.

Here, it is convenient to write W in this way :

W (f, g) := k_{η_{r}}^{ϵ_{r}} ∘ k_{η_{r−1}}^{ϵ_{r−1}} ∘ … k_{η_{2}}^{ϵ_{2}} ∘ k_{η_{1}}^{ϵ_{1}}  (8.342)
ϵ_{i}, η_{i} ∈ {+, −} , k_{+} := f , k_{−} := g , {η_{i} = η_{i+1}} ⇒ {ϵ_{i} = ϵ_{i+1}}

A “fundamental square” in ℝ^{+} ×ℝ^{+} is a square [a, b]×[a, b] that contains a “tentative portion” of the trivial graphs [z, f_{0}(z)], [f_{0}(z), z] and of the nontrivial graphs [z, g_{0}(z)], [g_{0}(z), z], with a full “cycle” of points Z_{i} duly located on these graphs and appearing in the proper succession.^{43}

42with, say, f known and g unknown.

h_{i} := k_{η_{i}}^{ϵ_{i}} ∘ ⋯ ∘ k_{η_{1}}^{ϵ_{1}}  (8.343)
Z_{1} := (h_{0}(z) , h_{1}(z)) = (z , h_{1}(z))
Z_{2} := (h_{1}(z) , h_{2}(z))
…
Z_{r−1} := (h_{r−2}(z) , h_{r−1}(z))
Z_{r} := (h_{r−1}(z) , h_{r}(z)) = (h_{r−1}(z) , z)

Functional continuation and periodic adjustment : For any (f̃_{0}, g̃_{0}) we may always construct such a square with an approximate (\overline{f}_{0} := T, \overline{g}_{0}), with \overline{f}_{0} := T and \overline{g}_{0} smooth, even analytic, except possibly at the extremal points z = a and z = b. Provided \overline{g}_{0} is close enough to a leading section of g̃_{0} (itself taken large enough to determine all the finitely many, discrete or continuous parameters on which the general twin solution (f, g) depends) there clearly exists a unique functional continuation of \overline{g}_{0} (that of \overline{f}_{0} is trivial) over the whole interval [b, +∞[, with \overline{g}_{0} everywhere smooth (or analytic) except possibly at a sequences of ‘images’ z_{3}, z_{4}, z_{5} … of z_{1}, z_{2} and with the consecutive images of the r points Z_{i} retaining their proper order inside the succesive images of the ‘fundamental square’.

An asymptotic analysis of \overline{g}_{0} at +∞ will reveal an oscillatory part interfering with the proper transserial part. But these parasitical oscillations may always be uniquely corrected by conjugation \overline{g}_{0} ↦ g_{0} = h^{−1} ∘ \overline{g}_{0} ∘ h with a suitable 1-periodic mapping h, ie one that commutes with the unit shift T. And not only does this unique h-conjugation spirit away the parasitical oscillations; it also has the automatic effect of restoring smoothness at the points z_{1}, z_{2}, z_{3} . . .. The pair (f_{0}, g_{0}) thus obtained may be regarded as the exact geometric counterpart, or ‘sum’, of the formal pair (f̃_{0}, g̃_{0}).

There exits, however, a significant difference depending on whether the formal pair (\overline{f}_{0}, \overline{g}_{0}) does or does not involve transexponentials E_{n}.

If it doesn’t, then (f_{0}, g_{0}) is determined absolutely and real-analytic on […, +∞[, though of course usually not at +∞.

43that order is always unambiguously determined by the formal twin (f̃_{0}, g̃_{0}), due to the full order that exists on transseries.

If it does, then (f_{0}, g_{0}) is determined relatively to a choice of E_{n} (and here the ‘indiscernibility principle’ comes into play) and probably (though this hasn’t been proved) real-analytic on […, +∞[ for a real-analytic choice of E_{n}. This is by no means a foregone conclusion, for with respect to general transserial functional equations, an analytic choice of E_{n} doesn’t guarantee the solution’s analyticity, but only its cohesiveness – which is a strong form of smoothness and quasi-analyticity : see [E5], [E6], [E7]. But the fact is that twin equations are a very special sub-class of composition equations, and enjoy a whole range of specific properties. So the jury is still out on that one.

8.5 The ordering of free or bound groups. Natural and exotic orders. Ultraexponential arbitration.

Natural and exotic orders on the composition group of transseries : Let 𝕀𝕋 denote the composition group of all (formal) identity-tangent transserial mappings f : z ↦ z + O(1) (z ∼ +∞) . Alongside its natural order ≺, the group 𝕀𝕋 admits a non-countable infinity of exotic orders ≺^{I,ϵ}.

Here is how to construct them. A transmonomial interval I := [A_{1}, A_{2}] or ]A_{1}, A_{2}] or [A_{1}, A_{2}[ or ]A_{1}, A_{2}[ bounded by two infinitesimal transmonomials A_{1} ≻ A_{2} is declared shift-invariant if for any transmonomial A ∈ I and any real shift t, the transseries A_{t}(z) := A(z + t) has as its leading term a transmonomial leadA_{t} that also lies in I. Not all intervals, of course, are shift-invariants, but those bounded by ordinary (negative) powers, for instance, are. Now, choose any partition I = I_{1} ∪ I_{2} …, finite or not, of the fully ordered set of all infinitesimal transmonomials (for the natural order) into shift-invariant intervals I_{i} and assign a sign ϵ_{i} ∈ {+, −} to each of these.

Then define on 𝕀𝕋 a binary relation ≺^{I,ϵ} as follows :

{g ≺^{I,ϵ} f} ⟺ {0 ≺ ϵ_{i} (f ∘ g^{−1} − id) if lead(f ∘ g^{−1} − id) ∈ I_{i}} (8.344)

The relation ≺^{I,ϵ} is clearly transitive and trivially stable under right composition by h ∈ 𝕀𝕋. It is also stable under left compostion since, due to the shiftinvariance of all I_{i}, the leading terms of f ∘g^{−1} −id and h∘f ∘g^{−1} ∘h^{−1} −id always fall within the same interval of the partition I. Each relation ≺^{I,ϵ} therefore defines an exotic order on 𝕀𝕋 that is compatible with the group structure as well as the natural topology.

Embedding-induced orders on free groups : Any isomorphism A_{i} ↦ f_{i} of the r-generator free group Gr(A) into a subgroup of the group 𝕀𝕋 endowed with its natural order ≺ or some exotic ≺^{I,ϵ}, clearly induces a total group order (with left- and right-stability) on Gr(A).

Similarly, for any system {f_{i}} of transserials twins or siblings , such an imbedding induces a total order on a suitable quotient Gr(A)/{W(A)}.

While there probably exist, on the free Gr(A) or its ‘bound’ quotients, other group orders than those obtainable in the above manner, this huge class of embedding-induced group orders is nonetheless quite interesting, if only because it points to three rather strange dichotomies :

– first, the dichotomy between natural/exotic orders, depending on whether the underlying embedding is into {𝕀𝕋, ≺} or {𝕀𝕋, ≺^{I,ϵ}}.

– second, the dichotomy between convergent/divergent orders, depending on whether all {f_{i}} can/cannot be chosen simultaneously convergent.

– third, the dichotomy between summable/non-summable orders, depending on whether all {f_{i}} can/cannot be chosen simultaneously summable^{44}.

Ultraexponential arbitration : In the case of twins or siblings {f_{i}}, the skilful addition to them of infinitesimal terms of type 1/E_{∞}, ie small beyond all transexponential orders, can always restore independence, and induce on Gr(A) group orders which sometimes can be obtained in no other way.

Thus, the order on the two-generator group Gr(A_{1}, A_{2}) corresponding to the independent pair {f_{1}(z) := E_{∞}(z) , f_{2}(z) := z + 1} or, what amounts to the same^{45}, to the pair {f̄_{1}(z) := z + 1 , f̄_{2}(z) := z + 1/E_{∞}(z)}, is specific to that (very) particular embedding, and capable of a remarkable combinatorial interpretation.

44this third dichotomy is quite distinct from the second one !

45indeed, both pairs {f_{1}, f_{2}} and {f̄_{1}, f̄_{2}} are conjugate under a “superfast” Ē_{∞}. Note, however, that replacing E_{∞} by Ē_{∞} in the above embedding would make no difference to the order induced on Gr(A_{1}, A_{2}). Indeed, although E_{∞} and Ē_{∞} are “distinguishable” when used jointly in the same relation – the latter is much faster growing than the former and all its finite iterates – they are nonetheless “undistinguishable” when occuring in isolation !

9 Conclusion : settled points, open questions.

Synoptic table : With the usual abbreviations :

{F, G} := F GF^{−1}G^{−1} , F_{n} := G^{n}F G^{−n} , G_{n} := F^{n}GF^{−n}

tangency defining relation  word  freedom analytic
order  W (F, G) = 1  complexity degree nature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(p, q) W_{1}(F, G) = {U_{1}, V_{1}} with not optim. ℂ divergent
U_{1} = {F {F {F, G}}}  biresurgent
V_{1} = {G{G, F }}  resummable
(p, q)  W_{2}(F, G) = {U_{1}, V_{1}} with  not optim. ℂ × ℚ divergent
U_{2} = {F {G{G{G, F }}}}  biresurgent
V_{2} = {G{F {G{F, G}}}}  resummable
(0, q)  W_{3}(F, G) = G_{1}^{−3}G_{0}^{4}G_{2}^{1}G_{0}^{4}G_{1}^{−3} (6, 15)  ℂ  analytic
(0, q)  W_{4}(F, G) = G_{0}^{8}G_{1}^{−10}G_{2}^{9}G_{3}^{−10}G_{4}^{8} (8, 45)  ℂ  divergent
(0, 0)  W_{5}(F, G) = W_{3}(F, {F, G})  (6, 30)  ℂ^{2}  analytic
(0, 0)  W_{6}(F, G) = W_{4}(F, {F, G})  (10, 30)  ℂ^{2}  divergent
(0^{∗}, 0^{∗}) F^{7} = G^{7} = (F G)^{7} = 1  ℂ^{∞}  analytic

Main settled points :

• Existence of numerous but sporadic types of formal twins.

• Generic convergence for non identity-tangent twins.

• Generic divergence-cum-resurgence for identity-tangent twins.

• Extension to ‘siblings’ : r+1 mappings constrained by r relations.

Main open questions :

• Are identity-tangent twins always divergent ?

• What is the arithmetic nature^{46} of their resurgence invariants ?

• What autarky relations do these verify^{47} ?

• Do all transserial formal twins possess a real-analytic (as opposed to merely cohesive) geometric realisation ?

Some avenues for exploration :

• Extend the Uker-filtration to the case of r-generator free algebras and higher dimensional differential representations.

• Calculate the dimensions that go with these filtrations.

• Investigate the higher-dimensional analogue of twins and siblings.

• Explore the potential^{48} of representations of free or ‘nearly free’ groups into groups of (one or many dimensional) germ mappings.

REFERENCES.

[Ce] D. Cerveau, Une liste de problèmes, in Ecuaciones Diferenciales, pp 455-460, Univ. Valladolid, 1997

[Co] S. D. Cohen, The group of translations and rational powers is free, Ph.D. thesis, Bowling Green State university, 19

[E1] J. Ecalle, Les fonctions résurgentes, Vol.1, Algèbres de fonctions résurgentes. Publ. Math. Orsay (1981).

[E2] J. Ecalle, Les fonctions résurgentes, Vol.2, Les fonctions résurgentes appliquées à l’itération. Publ. Math. Orsay (1981).

[E3] J. Ecalle, Les fonctions résurgentes, Vol.3, L’équation du pont et la classification analytique des objets locaux. Publ. Math. Orsay (1985).

[E4] J. Ecalle, The acceleration operators and their applications. Proc. Internat. Cong. Math., Kyoto, 1990, vol.2, Springer, Tokyo, 1991, p1249-1258.

46presumably transcendental

47when there are free continuous parameters.

48for the “word problem”, the description of all possible group orders, etc

[E5] J. Ecalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. Actual. Math., Hermann, Paris, 1992.

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