Entropy Rigidity of Uniformly Quasiconformal Anosov Flows

Yong Fang

Laboratoire de Mathématique d’Orsay, U.M.R. 8628 du C.N.R.S, Université Paris-sud, France (e-mail: fangyong1@yahoo.fr)

Abstract — Let ϕ_{t} be a C^{∞} topologically transitive uniformly quasiconformal Anosov flow, such that E^{+} ⊕ E^{−} is C^{∞} and the dimensions of E^{+} and E^{−} are at least 2. We show that if its Bowen-Margulis measure is in the Lebesgue class, then, up to finite covers, ϕ_{t} is C^{∞} flow equivalent to some algebraic model.

1. Introduction Conformal geometry is a classically and currently fascinating subjet,

which is meaningfully mixed together with hyperbolic dynamical systems under the impulsion of the classical [Su], [Ka] and [Yu] and the recent [Sa] and [K-Sa]. Let M be a C^{∞}-closed manifold. A C^{∞}-flow, ϕ_{t}, generated by a non-singular vector field X on M is called Anosov, if there exists a ϕ_{t}-invariant splitting of the tangent bundle

T M = ℝX ⊕ E^{+} ⊕ E^{−},

a Riemannian metric on M and two positive numbers a and b, such that

∀ u^{±} ∈ E^{±}, ∀ t > 0, ∥ Dϕ_{∓t}(u^{±}) ∥≤ a · e^{−bt} ∥ u^{±} ∥ .

Since M is compact, then the definition of an Anosov flow is independent of the Riemannian metric chosen. A metric is called Lyapunov, if with respect to it, the contant a above can be taken to be 1. Then for any Anosov flow, there exists always a Lyapunov metric (see [Sh]). E^{+} and E^{−} are called the unstable and stable distributions of ϕ_{t}. Denote E^{±} ⊕ℝX by E^{±,0}. Then E^{+,0} and E^{−,0} are called the weak unstable and weak stable distributions of ϕ_{t}. E^{±} and E^{±,0} all integrate to continuous foliations with C^{∞} leaves, denoted respectively by F^{±} and F^{±,0} (see [HK]). The corresponding leaves, passing by x, are denoted by W_{x}^{±} and W_{x}^{±,0}.

Define two functions on M × ℝ as follows,

K^{±}(x, t) := \frac{max\{∥ Dϕ_{t}(u) ∥ | u ∈ E_{x}^{±}, ∥ u ∥= 1\}}{min\{∥ Dϕ_{t}(u) ∥ | u ∈ E_{x}^{±}, ∥ u ∥= 1\}}.

Then we say that the Anosov flow, ϕ_{t}, is uniformly quasiconformal, if K^{+} and K^{−} are bounded. The corresponding notions for Anosov diffeomorphisms are defined similarly.

Recall that a flow, ψ_{t}, is called topologically transitive, if it has a dense orbit. It is called topologically mixing, if for ∀ U, V, nonempty open subsets of M, ∃ T > 0, such that ψ_{t}U ∩ V ≠ ∅, ∀ t ≥ T.

Among the dynamical invariants of an Anosov flow ϕ_{t}, the topological entropy is the most fundamental one, denoted by h_{top}(ϕ_{t}). Denote by M(ϕ_{t}) the set of ϕ_{t}-invariant probability measures. For ∀ ν ∈ M(ϕ_{t}), denote by h_{ν}(ϕ_{t}) the metrical entropy of (ϕ_{t}, ν) (see [HK]). It is classically shown (see [HK]) that if ϕ_{t} is topologically transitive, then there exists a unique ϕ_{t}-invariant probability, μ, such that

h_{μ}(ϕ_{t}) = h_{top}(ϕ_{t}) = sup_{ν∈M(ϕ_{t})}{h_{ν}(ϕ_{t})}.

This measure μ is called the Bowen-Margulis measure of ϕ_{t}. Our purpose is to prove the following

Theorem 1. Let ϕ_{t} be a C^{∞} topologically transitive uniformly quasiconformal Anosov flow on a closed manifold, such that E^{+} ⊕ E^{−} is C^{∞} and the dimensions of E^{+} and E^{−} are at least 2. If its Bowen-Margulis measure is in the Lebesgue class, then, up to a constant change of time scale and finite covers, ϕ_{t} is C^{∞} flow equivalent either to the suspension of a hyperbolic automorphism of a torus, or to the geodesic flow of a hyperbolic manifold.

Recently, P. Foulon proved an entropy rigidity theorem for three dimensional Anosov flows with C^{1} E^{+} ⊕ E^{−} (see [Fo]). In the case of dimension 3, an Anosov flow is certainly uniformly quasiconformal and the C^{1}-smoothness of E^{+} ⊕ E^{−} induces its C^{∞}-smoothness (see [Hu-K]). So Theorem 1 above is a higher dimensional generalization of this result of P. Foulon. Quite recently, B. Kalinin and V. Sadovskaya bring into the study of uniformly quasiconformal Anosov systems some new ideas (see [K-Sa] and [Sa]), among which is found our departing point.

In the section 2. below, we fix the terminology and precise several dynamical and geometrical lemmas. Then in the section 3, we prove Theorem 1. In the appendix, we prove a lemma about affine Anosov flows, which is used in the proof of Theorem 1.

2. Preliminaries. 2.1. Linearizations and smooth conformal structures.

In this subsection, we review and adapt some results of [Sa] to our situation. The starting point is the following proposition from [Sa].

Proposition 2.1.1. Let f be a diffeomorphism of a compact Riemannian manifold M, and let W be a continuous f-invariant foliation with C^{∞} leaves. Suppose that ∥ Df |_{T W} ∥< 1, and there exist C > 0 and ϵ > 0 such that for any x ∈ M and n ∈ ℕ,

∥ (Df^{n} |_{T_{x}W})^{−1} ∥ · ∥ Df^{n} |_{T_{x}W} ∥^{2} ≤ C(1 − ϵ)^{n}.

Then for any x ∈ M, there exists a C^{∞} diffeomorphism h_{x} : W (x) → T_{x}W such that (i). h_{fx} ∘ f = Df_{x} ∘ h_{x}, (ii). (Dh_{x})_{x} is the identity map, and (iii). h_{x} depends continuously on x in C^{∞} topology. In addition, the family h of maps h_{x} satisfying (i), (ii) and (iii) is unique.

Now let ϕ_{t} be a uniformly quasiconformal Anosov flow on a closed manifold M. Fix a Lyapunov metric on M. Then for ∀ s < 0,

∥ Dϕ_{s} |_{E^{+}}∥< 1.

Since the flow is uniformly quasiconformal, then for each negative number s, (ϕ_{s}, F^{+}) satisfies the conditions of the previous proposition. So for ∀ x ∈ M, there exists a C^{∞} diffeomorphism h_{x}^{+,s} : W_{x}^{+} → E_{x}^{+}, such that (i). h_{ϕ_{s}(x)}^{+,s} ∘ ϕ_{s} = D_{x}ϕ_{s} ∘ h_{x}^{+,s}, (ii). (Dh_{x}^{+,s})_{x} is the identity map, (iii). h_{x}^{+,s} depends continuously on x in the C^{∞} topology.

For ∀ m ∈ ℕ, we observe easily that \{h^{+, \frac{s}{m}} \}_{x∈M} satisfies also these three conditions with respect to ϕ_{s}. Then by the uniqueness of this family of maps for ϕ_{s}, we get

h_{x}^{+, \frac{s}{m}} = h_{x}^{+,s}, ∀ x ∈ M.

We deduce that for ∀ a ∈ ℚ and a < 0, h_{x}^{+,a} = h_{x}^{+,−1}, ∀ x ∈ M. Then by the condition (iii), we get

h_{ϕ_{t}(x)}^{+,−1} ∘ ϕ_{t} = (D_{x}ϕ_{t}) ∘ h_{x}^{+,−1}, ∀ x ∈ M, ∀ t ∈ ℝ.

Denote h_{x}^{+,−1} by h_{x}^{+}. Thus we have

h_{ϕ_{t}(x)}^{+} ∘ ϕ_{t} = (D_{x}ϕ_{t}) ∘ h_{x}^{+}, ∀ x ∈ M, ∀ t ∈ ℝ.

This family of C^{∞} maps {h_{x}^{+}}_{x∈M} is called the unstable linearization of E^{+}. Similarly, we get the stable linearization of E^{−}, denoted by {h_{x}^{−}}_{x∈M}.

For the sake of completeness, we precise the following

Lemma 2.1.1. Let ψ_{t} be a C^{∞} topologically transitive Anosov flow, then ψ_{t} is topologically mixing, iff ∀ x ∈ M, W_{x}^{+} is dense in M.

Proof. If ∃ x ∈ M, such that W_{x}^{+} is not dense in M, then by [Pl], E^{+}⊕E^{−} is integrable with compact C^{1} leaves. Then ψ_{t} is C^{1} flow equivalent to the suspension of a C^{1} Anosov diffeomorphism. So ψ_{t} is not topologically mixing.

Suppose that for ∀ x ∈ M, W_{x}^{+} is dense in M. Fix a Riemannian metric on M. For ∀ x ∈ M, r > 0, denote by B(x, r) and B^{+}(x, r) the balls of center x and rayon r in M and W_{x}^{+}. Take two open subsets U and V in M and a small ball B(y, ϵ) in V. Since each unstable leaf is dense in M and M is compact, then we can find R < +∞, such that

B^{+}(x, R) ∩ B(y, ϵ) ≠ ∅, ∀ x ∈ M.

Take a small disk B^{+}(x, δ) in U, then by the definition of an Anosov flow, ∃ T > 0, such that

ψ_{t}(B^{+}(x, δ)) ⊇ B^{+}(ψ_{t}(x), R), ∀ t ≥ T.

So ψ_{t}U ∩ V ≠ ∅, ∀ t ≥ T. □

Lemma 2.1.2. Let ϕ_{t} be a C^{∞} topologically transitive uniformly quasiconformal Anosov flow. If E^{+} ⊕ E^{−} is C^{∞} and the dimensions of E^{+} and E^{−} are at least 2, then E^{+} and E^{−} are C^{∞}.

Proof. If ϕ_{t} is topologically mixing, then by Theorems 1.3. and 1.4. of [Sa], E^{±,0} are C^{∞}. So if E^{+} ⊕ E^{−} is C^{∞}, then E^{±} are also C^{∞}.

If ϕ_{t} is not topologically mixing, then by Lemma 2.1.1, ∃ x ∈ M, such that W_{x}^{+} is not dense in M. Then by [Pl], E^{+} ⊕ E^{−} is integrable with compact leaves.

Take a leaf of E^{+} ⊕ E^{−}, Σ, and T > 0, such that ϕ_{T} (Σ) = Σ. Then ϕ_{T} is a topologically transitive uniformly quasiconformal Anosov diffeomorphism. Again by Theorems 1.3. and 1.4. of [Sa], the unstable and stable distributions of ϕ_{T} are C^{∞}. So E^{+} and E^{−} are C^{∞}. □

Lemma 2.1.3. Let ϕ_{t} be as in the lemma above, then ϕ_{t} preserves two C^{∞} conformal structures along F^{+} and F^{−}, denoted by τ^{+} and τ^{−}.

Proof. If ϕ_{t} is topologically mixing, then by Theorem 1.3. of [Sa], ϕ_{t} preserves a continuous conformal structure τ^{+} along F^{+}, which is C^{∞} along the leaves of F^{+}. Let y ∈ W_{x}^{−,0}, consider the weak stable holonomy map along the weak stable foliation

H_{x,y}^{−,0} : W_{x}^{+} → W_{y}^{+}
z ∈ W_{x}^{+} → W_{y}^{+} ∩ W_{z}^{−,0}.

By Lemma 2.1.2, E^{+} and E^{−} are C^{∞}. Then H_{x,y}^{−,0} depends smoothly on x and y. In the subsection 3.2. of [Sa], τ^{+} is proved to be invariant under the local weak stable holonomy maps. So τ^{+} is in fact C^{∞} on M. Similarly, we get a C^{∞} τ^{−} along F^{−}.

If ϕ_{t} is not topologically mixing, then we get the smooth τ^{+} and τ^{−} by similar arguments about the induced Anosov diffeomorphism on a leaf of E^{+} ⊕ E^{−}. □

Let ϕ_{t} be as in Lemma 2.1.2. For ∀ x ∈ M, we extend the conformal structure τ_{x}^{+} at 0 ∈ E_{x}^{+} to all other points of E_{x}^{+} via translations. We denote by σ_{x}^{+} this translation invariant conformal structure on E_{x}^{+}. Then by Lemma 3.1. of [Sa], for ∀ x ∈ M, h_{x}^{+} sends τ^{+} |_{W_{x}^{+}} to σ_{x}^{+}. So for ∀ y ∈ W_{x}^{+}, h_{x}^{+} ∘ (h_{y}^{+})^{−1} is a conformal diffeomorphism of (σ_{y}^{+}, E_{y}^{+}) to (σ_{x}^{+}, E_{x}^{+}). Since the dimension of E^{+} is at least 2, then h_{x}^{+}∘(h_{y}^{+})^{−1} is naturally an affine map. So if, by h_{x}^{+} and h_{y}^{+}, we push forward the canonical flat linear connections of E_{x}^{+} and E_{y}^{+} onto W_{x}^{+}, we get the same connection on W_{x}^{+}. Thus in this way, we get a well-defined continuous connection along F^{+}, denoted by ∇^{+} (see [Ka1] for some details about connections along a foliation).

By the condition (i) of the unstable linearization, ∇^{+} is seen to be ϕ_{t}- invariant. Similarly, we get a continuous ϕ_{t}-invariant connection along F^{−}, ∇^{−}. If the linearizations, {h_{x}^{±}}_{x∈M}, depend smoothly on x, then ∇^{+} and ∇^{−} are certainly C^{∞}.

2.2. Some smooth ergodic theory.

Let ϕ_{t} be a C^{∞}-flow on a closed manifold M and ν be a ϕ_{t}-invariant probability. If ϕ_{t} is ergodic with respect ν, then by the Multiplicative Ergodic Theorem of Oseledec, there exists a ν-conull ϕ_{t}-invariant subset Λ of M and a ϕ_{t}-invariant measurable (Lyapunov) decomposition of T M |_{Λ},

T M |_{Λ}= ⊕_{0≤i≤k}E_{i},

such that for ∀ u_{i} ∈ E_{i},

lim_{t→±∞} t^{−1} log ∥ Dϕ_{t}(u_{i}) ∥= χ_{i}.

Here E_{i} is called a Lyapunov subbundle and χ_{i} its Lyapunov exponent. E_{i} is also denoted by E_{χ_{i}}.

By definition, we say that the Lyapunov decomposition of ϕ_{t}, with respect to ν, is smooth, if there exists a ν-conull ϕ_{t}-invariant suset Λ_{1} of M and a C^{∞} decomposition of T M into smooth subbundles

T M = ⊕_{1≤i≤k}Ē_{i},

such that the Lyapunov decomposition is defined on Λ_{1} and

Ē_{i} |_{Λ_{1}}= E_{i} |_{Λ_{1}}, ∀ 1 ≤ i ≤ k.

If in addition the support of ν is M, then this C^{∞} decomposition of T M is certainly unique and ϕ_{t}-invariant. By convention, E_{a} := {0}, if a is not a Lyapunov exponent.

The following lemma is proved in [BFL1] (see Lemma 2.5. of [BFL1]).

Lemma 2.2.1. Under the notations above, if K is a measurable and bounded ϕ_{t}-invariant tensor of type (1, r), then on Λ, K(E_{i_{1}}, ⋯, E_{i_{r}}) ⊆ E_{χ_{i_{1}}+···+χ_{ir}} . Furthermore suppose that the Lyapunov decomposition is C^{∞} and the support of ν is M. Let ∇ be a ϕ_{t}-invariant C^{∞} connection on M, such that ∇Ē_{i} ⊆ Ē_{i}, ∀ 1 ≤ i ≤ k. If K is C^{∞}, then ∇K = 0, iff ∇Ē_{0}K = 0.

The following lemma is due to R. Feres and A. Katok (see [FK]).

Lemma 2.2.2. Under the notations above, if τ is a C^{∞} ϕ_{t}-invariant tensor of type (0, r) and ∑_{1≤l≤r} χ_{i_{l}} ≠ 0, then τ(E_{i_{1}}, ⋯, E_{i_{r}}) = 0.

Proof. Take the identity endomorphism of T M, Id, and construct the C^{∞} ϕ_{t}-invariant tensor τ ⊗ Id. Then

(τ ⊗ Id)(E_{i_{1}}, ⋯, E_{i_{r}}, E_{0}) = τ(E_{i_{1}}, ⋯, E_{i_{r}}) · Id(E_{0}) ⊆ E_{0}.

But by Lemma 2.2.1,

(τ ⊗ Id)(E_{i_{1}}, ⋯, E_{i_{r}}, E_{0}) ⊆ E_{χ_{i_{1}}+···+χ_{ir}} .

If ∑_{1≤l≤r} χ_{i_{l}} ≠ 0, then E_{0} ∩ E_{χ_{i_{1}}+···+χ_{ir}} = {0}. So τ(E_{i_{1}}, ⋯, E_{i_{r}}) = 0. □

If ϕ_{t} is an Anosov flow, then E_{0} = ℝX. If the stable and unstable distributions of ϕ_{t} are C^{∞}, then we get a smooth 1-form λ, such that

λ(E^{+} ⊕ E^{−}) = 0, λ(X) = 1.

We call λ the canonical 1-form of the flow, which is easily seen to be ϕ_{t}- invariant. If λ is a contact form, then ϕ_{t} is called a contact Anosov flow.

2.3. A geometric lemma

Let M and F be two C^{∞} manifolds. Suppose that the dimension of M is n. Denote by F^{1}(M) the frame bundle of M. If the general linear group, GL(n, ℝ), acts from left smoothly on F, then we get an associated fiber bundle, F^{1}M ⋊ F (see [K-No] for some details about fiber bundles). The C^{∞} sections of F^{1}M ⋊ F are called the geometric structures of type F and order 1. Given a C^{∞} linear connection ∇ on M, we get a horizontal distribution on F^{1}M ⋊ F, denoted by H_{F} (see [K-No]). Then a geometric structure, σ, is called ∇-parallel, if Dσ(T M) ⊆ H_{F}.

Let ϕ be a C^{∞} diffeomorphism of M, then ϕ acts on F^{1}(M) by its differential, Dϕ. For ∀ [α, a] ∈ F^{1}M ⋊ F (see [K-No]),

ϕ_{∗}[α, a] := [(Dϕ)(α), a].

Then ϕ acts naturally on the geometric structures and a geometric structure, σ, is called ϕ-invariant, if ϕ_{∗}σ = σ. Now we precise the following

Lemma 2.3.1. Let ∇ be a C^{∞} complete linear connection on a connected and simply-connected manifold, M. Let τ be a C^{∞} ∇-parallel geometric structure of order 1. If ∇T = 0 and ∇R = 0, then the group of C^{∞} affine transformations of M which preserve τ is a Lie group and acts transitively on M.

Proof. Note that the T and R above denote respectively the torsion and the curvature tensor of ∇. Denote by G_{τ} the group mentioned in the lemma. Then G_{τ} is a closed subgroup of the group of affine transformations. So G_{τ} is naturally a Lie group. Fix u_{0} ∈ F^{1}M and denote by P (u_{0}) the Holonomy subbundle of u_{0} (see [K-No]). Denote by G the group of affine diffeomorphisms of M which preserve P (u_{0}). Then by the assumptions, G is naturally a Lie group and acts transitively on M (see [K-No]). So we need only prove that g preserves τ, ∀ g ∈ G.

Suppose that τ is of type F. Take g ∈ G. For ∀ x ∈ M, ∃ u ∈ P (u_{0}) and a ∈ F, such that [u, a] = τ(x). Since g_{∗}(u) ∈ P (u_{0}), then there exists a piecewise smooth horizontal curve, u(t), in P (u_{0}), such that u(0) = u and u(1) = g_{∗}(u). Project u(t) to M and denote the resulting curve by γ. Then [u(t), a] gives a horizontal lift of γ in F^{1}M ⋊ F and [u(0), a] = τ(x). Since τ is ∇-parallel, then τ ∘ γ is also a horizontal lift of γ. We deduce that [u(t), a] = (τ ∘ γ)(t), ∀ t ∈ [0, 1], by the uniqueness of the horizontal lift beginning at a fixed point. Then we have

τ(g(x)) = τ(γ(1)) = [u(1), a]
= [g_{∗}(u), a] = g_{∗}[u, a] = g_{∗}(τ(x)).

So G ⊆ G_{τ}. We deduce that G_{τ} acts transitively on M □

3. The proof of Theorem 1. 3.1. Preparations.

Let ϕ_{t} be an Anosov flow on a closed manifold M, which satisfies the conditions of Theorem 1. Then by Lemma 2.1.2, E^{+} and E^{−} are C^{∞}. Up to finite covers, we suppose that M, E^{+} and E^{−} are both orientable. So if we denote by μ the Bowen -Margulis measure of ϕ_{t}, μ is given by the integration of a nowhere-vanishing C^{∞} volume form on M (see [L-S] and [So]).

At first, we suppose that ϕ_{t} is topologically mixing. Then G. A. Margulis proved (see [M]) that there exist two (unique up to scalars) families of measures, μ^{±,0}, supported by the leaves of F^{±,0}, such that

μ^{±,0} ∘ ϕ_{t} = e^{±ht}μ^{±,0},

where h denotes the topological entropy of ϕ_{t}. For ∀ x ∈ M and y ∈ W_{x}^{−}, consider the stable holonomy map along the stable foliation

H_{x,y}^{−} : W_{x}^{+,0} → W_{y}^{+,0}
z ∈ W_{x}^{+,0} → W_{y}^{+,0} ∩ W_{z}^{−}.

By [M], μ^{+,0} is invariant under the stable holonomy maps. Similarly, μ^{−,0} is invariant under the unstable holonomy maps (see also [HK]).

There exist also two families of measures, μ^{±}, supported by the leaves of F^{±}, such that

μ^{±} ∘ ϕ_{t} = e^{±ht}μ^{±}.

In addition, they are absolutely continuous with respect to the weak stable and weak unstable holonomy maps (see [HK]). Fix a C^{∞} Riemannian metric on M. Denote the induced Riemannian volume forms along F^{±} and F^{±,0} by ν^{±} and ν^{±,0}.

Lemma 3.1.1. Under the notations above, there exist C^{∞} positive functions on M, f^{±} and f^{±,0}, such that

μ^{±} = f^{±}ν^{±}, μ^{±,0} = f^{±,0}ν^{±,0}.

Proof. Let us prove at first that along each leaf of F^{+}, μ^{+} is absolutely continuous with respect to ν^{+}, denoted by μ^{+} ≪ ν^{+}.

Take a small ball B^{+}(x, δ) in W_{x}^{+} and A ⊆ B^{+}(x, δ), such that

ν^{+}(A) = 0.

By Lemma 2.1.2, E^{+} and E^{−} are C^{∞}. Then if δ is sufficiently small, the following map is a well defined local diffeomorphism for ∀ x ∈ M :

B^{−,0}(x, δ) × B^{+}(x, δ) → M,
(y, z) → B^{+}(y, 2δ) ∩ B^{−,0}(z, 2δ).

Set Ω := B^{−,0}(x, δ) × A, via the local diffeomorphisms above. Since E^{+} is C^{∞}, then for ∀ y ∈ B^{−,0}(x, δ), ν^{+}(y × A) = 0. Then by the Theorem of Fubini, Ω is of Lebesgue measure zero. Thus μ(A) = 0 by the assumption. But μ can be viewed as a product of μ^{+} and μ^{−,0} (see [Fo]), then

μ^{+}(y × A) = 0, μ^{−,0} − a.e.

Since μ^{+} is absolutely continuous with respect to the weak stable holonomy maps, then μ^{+}(A) = 0. So μ^{+} ≪ ν^{+}. Thus we get a measurable function f^{+} on M, such that f^{+} > 0 outside a Lebesgue measure zero set and

μ^{+} = f^{+}ν^{+}.

For ∀ (x, t) ∈ M × ℝ, define

f(x, t) := \frac{dν^{+}}{d(ν^{+} ∘ ϕ_{t})}(x).

Then it is easily seen that f is a multiplicative cocycle, positive and C^{∞} (see [Fo] and [Hu-K]). We have

f^{+}(ϕ_{t}(x)) = \frac{d(μ^{+} ∘ ϕ_{t})}{d(ν^{+} ∘ ϕ_{t})}(x)

= \frac{d(μ^{+} ∘ ϕ_{t})}{dμ^{+}} · \frac{μ^{+}}{ν^{+}} · \frac{dν^{+}}{d(ν^{+} ∘ ϕ_{t})} = e^{ht} · f^{+}(x) · f(x, t).

Then the Livsic cohomological theorem shows that f^{+} can be taken to be continuous and positive (see [Liv] and [Fo]). Since f is C^{∞}, then by the cocycle regularity theorem, f^{+} is in fact C^{∞} (see [LMM], [Hu-K] and [Wal]). Similarly we get the smooth and positive functions, f^{−} and f^{±,0}. □

Remark. The argument above is originally used by P. Foulon in the case of dimension 3 (see [Fo]).

Lemma 3.1.2. Under the notations above, h_{x}^{+} and h_{x}^{−} depend smoothly on x. Then in particular, ∇^{+} and ∇^{−} are C^{∞} on M.

Proof. Suppose that dimE^{+} = n (≥ 2). By Lemmas 2.1.2. and 2.1.3, E^{+} is C^{∞} and ϕ_{t} preserves τ^{+}, a C^{∞} conformal structure along F^{+}. By the previous lemma, μ^{+} is given by a family of C^{∞} volume forms along F^{+}. The volume of a frame of E^{+} is by definition the evaluation of μ^{+} on this frame. Then by claiming the τ^{+}-conformal frames of volume 1 to be orthonormal, we get a well defined C^{∞} Riemannian metric along F^{+}, denoted by g^{+}.

Take the leafwise Levi-Civita connections of g^{+}, denoted by ¯∇^{+}. Since τ^{+} is ϕ_{t}-invariant and

μ^{+} ∘ ϕ_{t} = e^{ht}μ^{+},

then

ϕ_{t}^{∗}g^{+} = e^{\frac{2h}{n} t}g^{+}.

We deduce that ¯∇^{+} is ϕ_{t}-invariant. For ∀ x ∈ M, define

h̄_{x}^{+} : E_{x}^{+} → W_{x}^{+},
u → exp ¯∇^{+}(u).

Because of the ϕ_{t}-invariance of ¯∇^{+}, we get

h̄_{ϕ_{t}(x)}^{+} ∘ Dϕ_{t} = ϕ_{t} ∘ h̄_{x}^{+}, ∀ x ∈ M, ∀ t ∈ ℝ.

Evidently, D_{x}(h̄_{x}^{+}) = Id and h̄_{x}^{+} depends smoothly on x.

Since g^{+} is certainly complete along each leaf of F^{+}, then h̄_{x}^{+} is surjective. Fix a Riemannian metric on M. Then by the compactness of M, there exists ϵ > 0, such that for ∀ x ∈ M, h̄_{x}^{+} |_{{u∈E_{x}^{+} | ∥u∥<ϵ}} is a C^{∞} diffeomorphism onto its image. If t ≫ 1, then ϕ_{−t} contracts E^{+} exponentially. Since we have in addition

h̄_{x}^{+} = ϕ_{t} ∘ h̄_{ϕ_{−t}(x)}^{+} ∘ Dϕ_{−t}, ∀ t > 0,

then h̄_{x}^{+} is in fact injective and nowhere singular. We deduce that h̄_{x}^{+} is a C^{∞} diffeomorphism, for ∀ x ∈ M. By the uniqueness of the unstable linearization (see Proposition 2.1.1), we get

h_{x}^{+} = (h̄_{x}^{+})^{−1}, ∀ x ∈ M.

So h_{x}^{+} depends smoothly on x. Similarly, we get the C^{∞} dependence of h_{x}^{−} on x. Then we deduce that ∇^{+} and ∇^{−} are C^{∞} on M (see the subsection 2.1). □

Remark. If ϕ_{t} is not topologically mixing and satisfies the conditions in Theorem 1, then by Lemma 2.1.1. and [Pl], E^{+} ⊕ E^{−} is integrable with compact leaves. Take a leaf Σ and T > 0, such that ϕ_{T} (Σ) = Σ. Then after some evident modifications, Lemmas 3.1.1. and 3.1.2. are also valid for (ϕ_{T} , Σ). Just as in the case of flow, we get two C^{∞} ϕ_{T} -invariant connections along F_{Σ}^{±}, denoted by ∇_{Σ}^{±} (see the subsection 2.1. and [K-Sa]). Then there exists on Σ a unique C^{∞} ϕ_{T} -invariant connection, ∇, such that

∇_{Y^{±}}Y^{∓} := P^{∓}[Y^{±}, Y^{∓}],
∇_{Y^{±}}Z^{±} := (∇_{Σ}^{±})_{Y^{±}}Z^{±}, ∀ Y^{±}, Z^{±} ⊆ E_{Σ}^{±},

where P_{Σ}^{±} denote the projections of T Σ onto E_{Σ}^{±}. By [BL], ϕ_{T} is C^{∞}- conjugate to a hyperbolic infranilautomorphism (see also [K-Sa]). Since ϕ_{T} is in addition uniformly quasiconformal, then up to finite covers, ϕ_{T} is C^{∞}-conjugate to a hyperbolic automorphism of a torus. So Theorem 1. is true in this case.

3.2. Homogeneity

Suppose that ϕ_{t} satisfies the conditions in Theorem 1. Because of the remark above, we suppose in addition that ϕ_{t} is topologically mixing. Since ϕ_{t} is uniformly quasiconformal, then with respect to μ, the Lyapunov exponents are {a^{−}, 0, a^{+}}, a^{−} < 0 < a^{+}. In particular, the Lyapunov decomposition is C^{∞}. Construct a C^{∞} connection ∇ on M, such that

∇X := 0, ∇E^{±} ⊆ E^{±},
∇_{Y^{±}}Z^{∓} := P^{∓}[Y^{±}, Z^{∓}], ∇_{Y^{±}}Z^{±} := (∇^{±})_{Y^{±}}Z^{±},

∇_{X}Y^{±} := [X, Y^{±}] + a^{±}Y^{±}, ∀ Y^{±}, Z^{±} ⊆ E^{±},

where P^{±} denote the projections of T M onto E^{±} with respect to the Anosov splitting. Then by a direct verification, ∇ is uniquely determined and ϕ_{t}- invariant.

Lemma 3.2.1. Under the notations above, dim E^{+} =dim E^{−}.

Proof. Since μ is given by a nowhere-vanishing ϕ_{t}-invariant volume form, then by Lemma 2.2.2,

dimE^{+} · a^{+} + dimE^{−} · a^{−} = 0.

We deduce that dimE^{+} = dimE^{−}, iff a^{+} + a^{−} = 0. Suppose on the contrary that a^{+} + a^{−} ≠ 0. Denote by T the torsion tensor of ∇. Then by Lemma 2.2.1,

T (E^{+}, E^{−}) ⊆ E_{a^{+}+a^{−}} = {0}.

By the definition of ∇, for arbitrary sections Y^{±} of E^{±}, we get

T (Y^{+}, Y^{−}) = dλ(Y^{+}, Y^{−}) · X,

where λ is the canonical 1-form defined in the subsection 2.2. We deduce that

dλ(E^{+}, E^{−}) = 0.

Since dλ is ϕ_{t}-invariant, then by the Anosov property,

dλ(X, E^{±}) = 0, dλ(E^{±}, E^{±}) = 0.

We deduce that dλ = 0. Then by Lemma A proved in the appendix, ϕ_{t} is C^{∞} flow equivalent to the suspension of a hyperbolic infranilautomorphism. So ϕ_{t} is not topologically mixing, which contradicts our assumption above. □

Up to a constant change of time scale, suppose in the following that dimE^{+} = n and a^{+} = 1. Set τ := (X, E^{±}, τ^{±}, μ), where μ is viewed as a C^{∞} volume form. Then by the subsections 2.1. and 2.3, τ is a C^{∞} ϕ_{t}-invariant geometric structure of order 1 on M.

Lemma 3.2.2. τ is ∇-parallel.

Proof. τ is in a natural sense the sum of the geometric structures, X, (E^{+}, τ^{+}), (E^{−}, τ^{−}) and μ. Then τ is ∇-parallel, iff these structures are parallel respectively.

Since ∇X = 0, then X is ∇-parallel. μ and ∇μ are both ϕ_{t}-invariant tensors. By Lemma 2.2.2, we get easily that ∇μ = 0, iff

(∇_{X}μ)(X, Y_{1}^{+}, ⋯, Y_{n}^{+}, Y_{1}^{−}, ⋯, Y_{n}^{−}) = 0,

where {Y_{i}^{±}}_{1≤i≤n} are arbitrary sections of E^{±}. By the definition of ∇,

(∇_{X}μ)(X, Y_{1}^{+}, ⋯, Y_{n}^{+}, Y_{1}^{−}, ⋯, Y_{n}^{−}) = (L_{X}μ)(⋯) = 0.

So μ is ∇-parallel.

Consider the structure (E^{+}, τ^{+}), denoted by σ^{+}. It is easily seen that σ^{+} is ∇-parallel, iff ∇E^{+} ⊆ E^{+} and the τ^{+}-conformal frames are preserved by the parallel transport of E^{+} along each piecewise smooth curve of M (see [K-No]). By the definition of ∇, the parallel transport of E^{+} along the orbits of ϕ_{t} is given by

u^{+} → e^{−t} · Dϕ_{t}(u^{+}).

Since τ^{+} is ϕ_{t}-invariant, then the τ^{+}-conformal frames are preserved along the orbits, i.e. σ^{+} ∘ ϕ_{t} is horizontal. So Dσ^{+}(X) ⊆ H, where H denotes the ∇-horizontal distribution of the corresponding fiber bundle.

Take a smooth curve γ, tangent to E^{+}. The restriction of ∇ to the leaves of F^{+} is ∇^{+}. On the leaf containing γ, ∇^{+} is equivalent to the canonical flat connection of a vector space. So the τ^{+}-conformal frames are preserved by the parallel transport along γ, i.e. σ^{+} ∘ γ is horizontal. We deduce that Dσ^{+}(E^{+}) ⊆ H.

Take another smooth curve γ, tangent to E^{−}. Then by the definition of ∇, the parallel transport along γ is given by the differentials of the weak stable holonomy maps. Since τ^{+} is invariant with respect to these maps (see [Sa]), then we deduce that Dσ^{+}(E^{−}) ⊆ H.

So we get Dσ^{+}(T M) ⊆ H, i.e. (E^{+}, τ^{+}) is ∇-parallel. Similarly, (E^{−}, τ^{−}) is also parallel. we deduce that τ is ∇-parallel. □

By Lemma 3.1.2, μ^{+,0} is given by a C^{∞} family of nowhere-vanishing volume forms along F^{+,0} (see the subsection 3.1.). Thus μ^{+,0} can be viewed as a C^{∞} nowhere-vanishing section of ∧^{n+1}(E^{+,0})^{∗}.

By claiming that μ^{+,0}(E^{−}, ⋯) := 0, μ^{+,0} can also be viewed as a C^{∞} (n + 1)-form on M. In any case, μ^{+,0} is a C^{∞} geometric structure of order 1 and in the following, we switch from one of these viewpoints to another without further precision.

Lemma 3.2.3. μ^{+,0} is ∇-parallel.

Proof. Since the Lyapunov exponents are {−1, 0, 1} and dimE^{+} = n, then by the entropy formula of Y. Pesin (see [Ma]), h_{top}(ϕ_{t}) = n. Then we have

μ^{+,0} ∘ ϕ_{t} = e^{nt}μ^{+,0}.

In fact, μ^{+,0} is ∇-parallel, iff it is parallel along all the curves, tangent to ℝX or E^{+} or E^{−} (see the proof of Lemma 3.2.2.).

By the definition of ∇, the parallel transport of E^{+} along an orbit of ϕ_{t} is given by

u^{+} → e^{−t} · Dϕ_{t}(u^{+}).

Since ∇X = 0, then X ∘ ϕ_{t} is parallel. Fix x ∈ M and a basis {u_{i}^{+}}_{1≤i≤n} of E_{x}^{+}. Define

S(t) := (X ∘ ϕ_{t}) ∧ e^{−t}Dϕ_{t}(u_{1}^{+}) ∧ ⋯ ∧ e^{−t}Dϕ_{t}(u_{n}^{+}).

Then S(t) is parallel along this ϕ_{t}-orbit, i.e. ∇_{∂_{t}}S(t) = 0. Since

μ^{+,0} ∘ ϕ_{t} = e^{nt}μ^{+,0},

then

(ϕ_{t})^{∗}μ^{+,0} = e^{nt}μ^{+,0},

i.e.

μ^{+,0}(ϕ_{t}(x)) = e^{nt} · (ϕ_{−t})^{∗}(μ^{+,0}(x)).

So with respect to the natural pairing of μ^{+,0} and S, we have

0 = ∂_{t} < μ^{+,0}(ϕ_{t}(x)), S(t) >
=< ∇_{∂_{t}}μ^{+,0}, S > + < μ^{+,0}, ∇_{∂_{t}}S >
=< ∇_{∂_{t}}μ^{+,0}, S > .

So ∇_{∂_{t}}μ^{+,0} = 0, i.e. μ^{+,0} is parallel along the orbits of ϕ_{t}.

Take a smooth curve γ, tangent to E^{−} and beginning at x. Since μ^{+,0} is invariant under the stable holonomy maps, then we get

μ^{+,0}(γ(t)) = (H_{x,γ(t)}^{−})_{∗}(μ^{+,0}(x)).

By the definition of ∇, for ∀ u^{+} ∈ E_{x}^{+}, the parallel transport of u^{+} along γ is obtained by the differentials of the weak stable holonomy maps,

u^{+} → (DH_{x,γ(t)}^{−,0})(u^{+}).

Take a small curve l, tangent to E^{+} and with u^{+} as the tangent vector at 0. Then for ∀ s ≪ 1, H_{x,γ(t)}^{−}(l(s)) and H_{x,γ(t)}^{−,0}(l(s)) are contained in W_{γ(t)}^{+,0} ∩ W_{l(s)}^{−,0}. Then for ϵ ≪ 1, we can find a smooth function b : [0, ϵ] → ℝ, such that b(0) = 0 and

H_{x,γ(t)}^{−}(l(s)) = ϕ_{b(s)}(H_{x,γ(t)}^{−,0}(l(s))), ∀ s ∈ [0, ϵ].

By differenting the relation above with respect to s at 0, we get a number a(t), such that

DH_{x,γ(t)}^{−}(u^{+}) = DH_{x,γ(t)}^{−,0}(u^{+}) + a(t) · X(γ(t)).

Take a basis of E_{x}^{+} as above, {u_{1}^{+}, ⋯, u_{n}^{+}}. Define

S(t) := (DH_{x,γ(t)}^{−})(X_{x} ∧ u_{1}^{+} ∧ ⋯ ∧ u_{n}^{+}).

Then by the relation above, we have

S(t) = X(γ(t)) ∧ (DH_{x,γ(t)}^{−,0})(u_{1}^{+}) ∧ ⋯ ∧ (DH_{x,γ(t)}^{−,0})(u_{n}^{+}).

We deduce that

∇_{∂_{t}}S = 0.

Then as above, we have

0 = ∂_{t} < μ^{+,0}(t), S(t) >=< ∇_{∂_{t}}μ^{+,0}, S(t) > .

So μ^{+,0} is parallel along γ.

Now take a curve γ, tangent to E^{+} and beginning at x. On the line bundle ∧^{n+1}(E^{+,0})^{∗}, ∇ induces naturally a connection ∇_{1}. Then certainly, μ^{+,0} is parallel, iff ∇_{1}μ^{+,0} = 0.

Along each curve l, denote by P_{s_{1},s_{2}}^{l} the parallel transport of ∇_{1} from l(s_{1}) to l(s_{2}). Denote by Ω^{+,0} the curvature form of ∇_{1}. Then Ω^{+,0} is a ϕ_{t}-invariant 2-form. By the Anosov property of ϕ_{t}, we get

Ω^{+,0}(X, E^{±}) = 0, Ω^{+,0}(E^{±}, E^{±}) = 0.

So the restriction ∇_{1} |_{W_{x}^{+,0}} is flat. We deduce that if two curves are homotopic with fixed endpoints in W_{x}^{+,0}, then their parallel transports are the same. For ∀ y ∈ M, denote by O_{y} the ϕ_{t}-orbit of y. Then for ∀ s > 0, we have

c(t) := \frac{P_{0,t}^{γ}(μ^{+,0}(x))}{μ^{+,0}(γ(t))}

= \frac{P_{0,−s}^{O_{γ(t)}} ∘ P_{0,t}^{γ}(μ^{+,0}(x))}{P_{0,−s}^{O_{γ(t)}}(μ^{+,0}(γ(t))}

= \frac{P_{0, t}^{ϕ_{−s}∘γ}(μ^{+,0}(ϕ_{−s}(x))}{μ^{+,0}(ϕ_{−s}(γ(t))},

where we have used that μ^{+,0} is parallel along the orbits of ϕ_{t}. If s → +∞, then the length of ϕ_{−s}∘γ goes to zero. By the compactness of M, if s → +∞, then c(t) goes to 1. So c(t) = 1, i.e. μ^{+,0} is parallel along γ. We conclude that μ^{+,0} is ∇-parallel. □

View μ^{+,0} as a C^{∞} (n + 1)-form and define ω^{+} := i_{X}μ^{+,0}. Since X and μ^{+,0} are ∇-parallel, then ω^{+} is also ∇-parallel. Similarly, we get a ∇-parallel n-form, ω^{−}.

Set σ := (τ, ω^{+}, ω^{−}), then σ is a C^{∞} ∇-parallel geometric structure of order 1 on M. Let M̃ be the universal covering space of M. Denote by σ̃ and ˜∇ the lifts of σ and ∇ to M̃.

Lemma 3.2.4. The group of ˜∇-affine transformations of M̃, which preserve σ̃, is a Lie group and acts transitively on M̃.

Proof. Let us prove at first that ∇T = 0 and ∇R = 0. Suppose that K is a C^{∞} ϕ_{t}-invariant tensor of type (1, k). Take arbitrarily the Lyapunov exponents {χ_{1}, ⋯, χ_{k}} and C^{∞} vector fields {Y_{1}, ⋯, Y_{k}}, such that

Y_{i} ⊆ E_{χ_{i}}, ∀ 1 ≤ i ≤ k.

By the definition of ∇ and Lemma 2.2.1, we have

(∇_{X}K)(Y_{1}, ⋯, Y_{k}) = ∇_{X}K(Y_{1}, ⋯, Y_{k}) − ∑_{1≤i≤k} K(Y_{1}, ⋯, ∇_{X}Y_{i}, ⋯, Y_{k})

= [X, K(Y_{1}, ⋯, Y_{k})] + ( ∑_{1≤i≤k} χ_{i})K(Y_{1}, ⋯, Y_{k}) − K([X, Y_{1}] + χ_{1}Y_{1}, ⋯) ⋯

= [X, K(Y_{1}, ⋯, Y_{k})] − ∑_{1≤i≤k} K(Y_{1}, ⋯, [X, Y_{i}], ⋯, Y_{k})

= (L_{X}K)(Y_{1}, ⋯, Y_{k}) = 0.

So ∇_{X}K = 0. Then by Lemma 2.2.1, ∇K = 0. In particular, we get

∇T = 0, ∇R = 0.

By Lemma 2.2.3. of [BFL2], the ∇-geodesics tangent to E^{+} or E^{−} are defined on ℝ. Then by Lemma A in [Fa], ∇ is complete. So ˜∇ is also complete. We conclude by Lemmas 2.3.1. and 3.2.1. □

Denote by G the Lie group in the previous lemma. Then G can be viewed as the symmetry group of our dynamical system. Fix a point x ∈ M̃ and denote by H the isotropy subgroup of x. Then G/H ≅ M̃. Denote by Γ the fundamental group of M, then Γ is contained in G as a discrete subgroup.

The isotropy representation of H is defined as follows,

i : H → GL(T_{x}M̃),
h → D_{x}h.

Since H preserves the connection, ˜∇, then i is injective.

Lemma 3.2.5. H has only finitely many connected components and i(H) is closed in GL(T_{x}M̃).

Proof. For all the terminology used below, our reference is [C-Q]. Since M̃ ≅ G/H, then M̃ is equipped naturally a real analytic structure, which makes the action of G on M̃ real analytic. Since σ̃ is invariant under the action of G, then σ̃ is also real analytic. From the definition of ∇, we observe easily than each isometry of σ̃ preserves ˜∇. Thus G = Isom(σ̃), where Isom(σ̃) denotes the isometry group of σ̃. For ∀ u ∈ 𝔤_{G}, we have an induced C^{∞} vector field on M̃,

Y^{u} : M̃ → T M̃

a → \frac{d}{dt} |_{t=0} exp(−tu)a.

Denote by Kill(σ̃) the set of the Killing fields of σ̃, then for ∀ u ∈ 𝔤_{G}, Y^{u} ∈ Kill(σ̃). Define

ψ : 𝔤_{G} → Kill(σ̃)
u → Y^{u}.

It is easily seen that ψ is an injective Lie algebra homomorphism (see for example [Va]). Let λ be the canonical 1-form of ϕ_{t} (see the subsection 2.2). Then λ^{2} gives a fiber metric of the vector bundle, ℝX. Take g^{+} and g^{−} as in Lemma 3.1.2. Then we get a C^{∞} Riemannian metric on M,

g := λ^{2} ⊕ g^{+} ⊕ g^{−}.

Since each local isometry of σ̃ preserves g̃, then each Killing field of σ̃ is also Killing for ˜∇_{g}, the Levi-Civita connection of g̃. Since M is compact, then ˜∇_{g} is complete. So each Killing field of σ̃ is complete (see [K-No]). We deduce that ψ is in fact an isomorphism. In addition, by the canonical existence of g, σ is seen to be a rigid geometric structure of order 1 (see [D-G] ch.5).

Suppose that r >> 1, then certainly the following map is an injective Lie group homomorphism,

i^{1+r} : H → Aut^{1+r}(σ̃, x)
h → j_{x}^{1+r}h.

By ([C-Q] 4), Kill_{0}^{1+r}(σ̃, x) can be viewed as the Lie algebra of Aut^{1+r}(σ̃, x). Similarly, Kill_{0}(σ̃, x) can be viewed as the Lie algebra of H under ψ. Then for ∀ Y ∈ Kill_{0}(σ̃, x), we have

D(i^{1+r})(Y) = j_{x}^{1+r}Y.

Since i^{1+r} is injective, then D(i^{1+r}) is also injective. By [Am] and Proposition 7.3. of [C-Q], each element of Kill_{0}^{1+r}(σ̃, x) can be uniquely extended to a global Killing field of σ̃. We deduce that D(i^{1+r}) is a Lie algebra isomorphism. So i^{1+r}(H) is an open (then closed) subgroup of Aut^{1+r}(σ̃, x) and

i(H_{e}) = (Aut^{1+r}(σ̃, x))_{e}.

Since Aut^{1+r}(σ̃, x) is a real algebraic subgroup of GL(T_{x}M̃), then Aut^{1+r}(σ̃, x) has only finitely many connected components. We deduce that H has also finitely many connected components. Define (see [C-Q])

π_{1}^{1+r} : Aut^{1+r}(σ̃, x) → GL(T_{x}M̃)(≅ D_{x}^{(1)}(M̃))
j_{x}^{1+r}f → j_{x}^{1}f.

Since Aut^{1+r}(σ̃, x) is real algebraic, then π_{1}^{1+r}(Aut^{1+r}(σ̃, x)) is closed in GL(T_{x}M̃) (see [Bo] ch.2).

Since i = π_{1}^{1+r} ∘ i^{1+r} and i is injective, then under i |_{H_{e}},

H_{e} ≅ [π_{1}^{1+r}(Aut^{1+r}(σ̃, x)]_{e}.

So i(H) is an open subgroup of π^{1+r}(Aut^{1+r}(σ̃, x)). We deduce that i(H) is closed in GL(T_{x}M̃). □

From the lemma above, we deduce that G has also finitely many connected components. So up to finite covers, we can suppose that G and H are connected and the fundamental group of M is contained in G.

Take a basis of T_{x}M̃, {X̃_{x}, u_{1}^{+}, ⋯, u_{n}^{+}, u_{1}^{−}, ⋯, u_{n}^{−}}. Since for ∀ h ∈ H, h preserves σ_{x}, then in this basis,

i(H) ⊆ {1} × SO(n) × SO(n).

We deduce that H is a compact Lie subgroup of G.

3.3. Symmetric Anosov flows.

In this subsection, the arguments are based on [To1] and [To2]. Recall at first the following definitions (see [To2]),

Definition. Let ψ_{t} be a C^{∞} flow on M, then a Lie transformation group G of M is called a symmetric group of (M, ψ_{t}), if G centralizes {ψ_{t}} in Diff(M) and the isotropy subgroups are compact in G.

The flow ψ_{t} is called symmetric, if there exists a normal covering space M̄ of M, such that the group of deck transformations is contained as a discrete subgroup in an effective and transitive symmetric group of (M̄, ψ̄_{t}).

Compare to [To2], we have added into the definition the effectiveness of the symmetric group action, which makes no essential difference. Let ψ_{t} be a symmetric flow as in the definition above. Denote the symmetric group on M̄ by G and the isotropy subgroup by K. Their lie algebras are denoted by 𝔤 and 𝔤_{K}. Denote the deck group by Γ (⊆ G), thus M ≅ Γ⧹G/K. By Proposition 1. of [To2], ∃ α ∈ 𝔤, such that [α, 𝔤_{K}] ≡ 0 and

ψ_{t}(ΓgK) = Γ(g · exptα)K, ∀ t ∈ ℝ.

Then by the fundamental estimation in Theorem 1. of [To1], we get easily

Proposition 3.3.1. Let ψ_{t} be a symmetric flow as above, then ψ_{t} is Anosov iff the kernel of adα is 𝔤_{K} + ℝα and adα has no nonzero imaginary eigenvalues. If ψ_{t} is in addition uniformly quasiconformal, then ℜ(Spec(adα)) has only three elements.

By combining Theorems 5. and 6. of [To2], we get the following

Proposition 3.3.2. Let ψ_{t} be a symmetric Anosov flow on a closed manifold. If ψ_{t} is neither a suspension nor a contact flow, then a lift of ψ_{t}, ψ̄_{t}, can be constructed as follows.

Define G := N ⋊(Spin(n, 1)×K_{1} ×⋯×K_{p}), where N is a vector group of positive dimension and K_{1}, ⋯, K_{p} are compact, simply connected and almost simple Lie groups. Let 𝔨′ ⊕ 𝔭 be a Cartan decomposition of 𝔰𝔬(n, 1) and α be a non-zero element of 𝔭. Let 𝔨 be the centralizer of α in 𝔨′ and K the connected Lie subgroup of G with Lie algebra 𝔨 ⊕ 𝔤_{K_{1}} ⊕ ⋯ ⊕ 𝔤_{K_{p}}. Then we have Ker(adα) = 𝔤_{K} ⊕ ℝα and

ψ̄_{t}(gK) = (g · exptα)K, ∀ t ∈ ℝ, ∀g ∈ G.

Now suppose that ϕ_{t} satisfies the conditions in Theorem 1. and is topologically mixing. Then by Subsection 3.2, ϕ_{t} turns out to be a symmetric Anosov flow.

By Proposition 3. of [To2] and the previous lemma, it is easily seen that up to finite covers, each contact symmetric Anosov flow must be C^{∞} flow equivalent to the geodesic flow of a locally symmetric space of rank 1. So if ϕ_{t} is contact, then up to finite covers, it is C^{∞} flow equivalent to the geodesic flow of a locally symmetric space of rank 1. Since ϕ_{t} is in addition uniformly quasiconformal, then the locally symmetric space in question must have constant negative curvature. Now we finish the proof of Theorem 1. by proving

Lemma 3.3.1. Let ϕ_{t} be as above, then ϕ_{t} must be contact.

Proof. Suppose on the contrary that ϕ_{t} is not contact. Since ϕ_{t} is topologically mixing, then it is neither a suspension. Thus using the notations of Proposition 3.3.2, a lift of ϕ_{t} is given by

G/K →^{ϕ̄_{t}} G/K

gK → (g · exptα)K,

such that Ker(adα) = 𝔤_{K} ⊕ ℝα.

Since α ∈ 𝔭 and 𝔰𝔬(n, 1) is of rank 1, then there exists a > 0 and X_{±} ∈ 𝔰𝔬(n, 1), such that

[a · α, X_{±}] = ±2X_{±}, [X_{+}, X_{−}] = −a · α.

Denote a · α again by α, i.e. consider the flow given by a · α. Denote by 𝔤_{α} the Lie subalgebra generated by {α, X_{+}, X_{−}}, then

𝔤_{α} ≅ 𝔰𝔩(2, ℝ).

Recall that G = N ⋊ (Spin(n, 1) × K_{1} × ⋯ × K_{p}), where N is a vector group of positive dimension. By identifying N with its Lie algebra, we get from this semidirect product a linear representation of 𝔰𝔬(n, 1) on N, thus a representation of 𝔤_{α}. Then we get a 𝔰𝔩(2, ℝ)-module (see [Bo1] 3).

Now by Proposition 2. of ([Bo2] ch.VIII, 1.2), there exists a nonzero vector e in N and m ∈ ℤ^{+} ∪ {0}, such that

α(e) = m · e.

Since Ker(adα) = 𝔤_{K}⊕ℝα, then ker(adα)∩N = {0} (see Proposition 3.3.2). So m ≠ 0. By Proposition 3.3.1, ℜ(spec(adα)) has only three elements. In addition, X_{+} is taken such that

[α, X_{+}] = 2X_{+}.

We deduce that m = 2. Define e_{1} = −X_{−}(e), then again by Proposition 2. of ([Bo2] ch.VIII, 1.2), we get

e_{1} ≠ 0, α(e_{1}) = 0,

which contradicts to Ker(adα) ∩ N = {0}. □

Remark. It is easily seen that each symmetric Anosov flow preserves a volume form and its Bowen-Margulis measure is in the Lebesgue measure class (see [Bow]).

4. Appendix.

Lemma A. Let ϕ_{t} be a C^{∞} topologically transitive Anosov flow with smooth distributions defined on a closed manifold M. If it preserves a C^{∞} linear connection and E^{+} ⊕ E^{−} is integrable, then ϕ_{t} is C^{∞} flow equivalent to the suspension of a hyperbolic infranilautomorphism.

Proof. Remark at first that E^{+} ⊕ E^{−} is integrable, iff dλ = 0, where λ is the canonical 1-form of ϕ_{t}.

Since E^{+} ⊕ E^{−} is integrable, then by Theorem 3.1. of [Pl], ϕ_{t} admits a C^{∞} global section Σ. Denote by r and ϕ the first return time function and the Poincaré map of Σ. Then ϕ is an Anosov diffeomorphism with smooth stable and unstable distributions, denoted by E_{Σ}^{−} and E_{Σ}^{+} (see [Fa]). In addition,

E_{Σ}^{±} = T Σ ∩ E^{±,0}.

Denote by ∇ the connection preserved by ϕ_{t}. Then the following connection,

¯∇ := P^{+} · ∇ · P^{+} + P^{−} · ∇ · P^{−} + P^{X} · ∇ · P^{X},

is also preserved by ϕ_{t}. So without any loss of generality, we can suppose that ∇E^{±} ⊆ E^{±}. Denote by ∇^{+} the restriction of ∇ to E^{+}. Then ∇^{+} is a ϕ_{t}-invariant C^{∞} connection along F^{+}. Now we want to construct a ϕ-invariant connection along F_{Σ}^{+}.

Suppose that dimE^{+} = k. Then for ∀ x ∈ M, W_{x}^{+} is C^{∞} diffeomorphic either to ℝ^{k} ×𝕊^{1} or to ℝ^{k+1}, depending whether or not it contains a periodic orbit. In addition, W_{x}^{+,0} contains at most one periodic orbit. Take x ∈ Σ, then W_{Σ,x}^{+} ⊆ W_{x}^{+,0}. Thus for ∀ y ∈ W_{Σ,x}^{+}, ∃ t ∈ ℝ, such that

ϕ_{t}(y) ∈ W_{x}^{+}.

If W_{x}^{+,0} contains no periodic orbit, then this number, t, is unique for ∀ y ∈ W_{Σ,x}^{+}. Suppose that W_{x}^{+,0} contains a periodic orbit of period T. If ϕ_{t}(y) ∈ W_{x}^{+}, then ϕ_{t+kT} (y) ∈ W_{x}^{+}, ∀ k ∈ ℤ. Inversely, if ϕ_{t_{1}}(y) ∈ W_{x}^{+} and ϕ_{t_{2}}(y) ∈ W_{x}^{+}, then ∃ k ∈ ℤ, such that t_{1} − t_{2} = kT. So by associating t to y, we get a well defined map of W_{Σ,x}^{+} to ℝ/T ℤ.

Thus in any case, for ∀ x ∈ Σ, there always exists a unique C^{∞} function

ζ_{x} : W_{Σ,x}^{+} → ℝ,

such that

ζ_{x}(x) = 0, ϕ_{ζ_{x}(y)}(y) ∈ W_{x}^{+}, ∀ y ∈ W_{Σ,x}^{+}.

Define Θ_{x} := ϕ_{ζ_{x}(.)}(.). Then there exists evidently a neighborhood of x, U_{Σ,x}, such that Θ_{x} |_{U_{Σ,x}} is a diffeomorphism onto its image. Fix arbitrarily z in U_{Σ,x} ∩ U_{Σ,y}. Define α := ζ_{y}(z) − ζ_{x}(z), then ϕ_{α}W_{x}^{+} = W_{y}^{+}. Thus ∃ k ∈ ℤ, such that

ζ_{y} = ζ_{x} + α + k · T,

where T is taken to be zero, if W_{y}^{+,0} contains no periodic orbit. We deduce that in a neighborhood of z,

Θ_{y} ∘ Θ_{x}^{−1} = ϕ_{α+k·T} .

Since ∇^{+} is ϕ_{t}-invariant, then we can pull back ∇^{+} by {Θ_{x} |_{U_{Σ,x}}}_{x∈Σ} to get a well-defined C^{∞} connection along F_{Σ}^{+}, denoted by ∇_{Σ}^{+}.

For ∀ x ∈ Σ, define

η := r(x) + ζ_{x} − r.

Then for ∀ y ∈ W_{Σ,x}^{+}, we have

ϕ_{η(y)}(ϕ(y)) = ϕ_{r(x)+ζ_{x}(y)}(y) ∈ W_{ϕ(x)}^{+}.

Since ϕW_{Σ,x}^{+} = W_{Σ,ϕ(x)}^{+} and η(x) = 0, then by the uniqueness of ζ_{ϕ(x)}, we get

ζ_{ϕ(x)} = η ∘ ϕ^{−1}.

i.e.

ζ_{ϕ(x)} ∘ ϕ = r(x) − r + ζ_{x}.

We deduce that

Θ_{ϕ(x)} ∘ ϕ = ϕ_{r(x)} ∘ Θ_{x}.

So ∇_{Σ}^{+} is ϕ-invariant. Similarly, we get ∇_{Σ}^{−}, a ϕ-invariant connection along F_{Σ}^{−}.

Thus we can construct a C^{∞} ϕ-invariant connection ∇_{Σ} on Σ, such that

(∇_{Σ})_{Y_{Σ}^{±}}Z_{Σ}^{∓} := P_{Σ}^{∓}[Y_{Σ}^{±}, Z_{Σ}^{∓}],
(∇_{Σ})_{Y_{Σ}^{±}}Z_{Σ}^{±} := (∇_{Σ}^{±})_{Y_{Σ}^{±}}Z_{Σ}^{±}.

Since ϕ_{t} is topologically transitive, then so is ϕ. By [BL], ϕ is seen to be C^{∞} conjugate to a hyperbolic infranilautomorphism. Then by Corollary 3.5. of [Pl], the leaves of E^{+} ⊕ E^{−} are compact. So we can take a leaf of E^{+} ⊕ E^{−} as Σ and the lemma follows. □

Acknowledgements. The author would like to thank his thesis advisors, P. Pansu and P. Foulon, for the discussions and help. He would like also to thank F. Labourie for clarifying discussions and A. Katok for mentionning kindly the papers, [K-Sa] and [Sa].

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