Anosov Flows Research Articles

A remark about hyperbolic infranilautomorphisms

We show that any exact 2-form, preserved by a hyperbolic infranilautomorphism, must be zero. We then deduce two propositions about geometric Anosov flows and the time change of suspensions. To cite this article: Y. Fang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Counting geodesics which are optimal in homology

On a surface of negative curvature, we study the distribution of closed geodesics which are not far from minimizing length in their homology classes. These geodesics do not have the statistical behaviour obtained by Margulis for ‘generic’ geodesics and later extended by Lalley to geodesics satisfying certain homological constraints. We cannot use the usual techniques of ‘thermodynamic formalism for Anosov flows’, but we study the problem using ideas from Aubry–Mather theory, as well as some facts of low-dimensional topology.

Geometric Anosov flows of dimension 5

We show that for a smooth Anosov flow on a closed five dimensional manifold, if it has C ∞ Anosov splitting and preserves a C ∞ pseudo-Riemannian metric, then up to a special time change and finite covers, it is C ∞ flow equivalent either to the suspension of a symplectic hyperbolic automorphism of T 4 , or to the geodesic flow on a three dimensional hyperbolic manifold. To cite this article: Y. Fang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Zygmund strong foliations

We show that for a volume-preserving Anosov flow on a 3-manifold the strong stable and unstable foliations are Zygmund-regular. We also exhibit an obstruction to higher regularity, which admits a direct geometric interpretation. Vanishing of this obstruction implies high smoothness of the joint strong subbundle and that the flow is either a suspension or a contact flow.

Pseudo-Anosov Flows and Incompressible Tori

We study incompressible tori in 3-manifolds supporting pseudo-Anosov flows and more generally Z ⊕ Z subgroups of the fundamental group of such a manifold. If no element in this subgroup can be represented by a closed orbit of the pseudo-Anosov flow, we prove that the flow is topologically conjugate to a suspension of an Anosov diffeomorphism of the torus. In particular it is non singular and is an Anosov flow. It follows that either a pseudo-Anosov flow is topologically conjugate to a suspension Anosov flow, or any immersed incompressible torus can be realized as a free homotopy from a closed orbit of the flow to itself. The key tool is an analysis of group actions on non-Hausdorff trees, also known as R-order trees – we produce an invariant axis in the free action case. An application of these results is the following: suppose the manifold has an R-covered foliation transverse to a pseudo-Anosov flow. If the flow is not an R-covered Anosov flow, then it follows that the manifold is atoroidal.

Axiom A flows with a transverse torus

Let X X be an Axiom A flow with a transverse torus T T exhibiting a unique orbit O O that does not intersect T T . Suppose that there is no null-homotopic closed curve in T T contained in either the stable or unstable set of O O . Then we show that X X has either an attracting periodic orbit or a repelling periodic orbit or is transitive. In particular, an Anosov flow with a transverse torus is transitive if it has a unique periodic orbit that does not intersect the torus.

On the approximation of time one maps of Anosov flows by Axiom A diffeomorphisms

We prove that if 𝒻1 is the time one map of a transitive and codimension one Anosov flow φ and it is C1-approximated by Axiom A diffeomorphisms satisfying a property called P, then the flow is topologically conjugated to the suspension of a codimension one Anosov diffeomorphism. A diffeomorphism 𝒻 satisfies property P if for every periodic point in M the number of periodic points in a fundamental domain of its central manifold is constant.

Ergodicity of stable manifolds for nilpotent extensions of Anosov flows

In this paper we show ergodicity of the strong stable foliations for nilpotent extensions of transitive Anosov flows with respect to the lift of the Gibbs measure for any Holder continuous function.

Plane affine geometry and Anosov flows

We study Anosov flows on closed 3-manifolds. We define the notion of Anosov flows with the topological contact property (abreviation TCP Anosov flows): typical examples of TCP Anosov flows are contact Anosov flows, i.e. flows preserving a contact 1-form. We show that TCP Anosov flows are R -covered. The main tool is the study of the leaf spaces of lifted strong stable foliations: we exhibit on these leaf spaces a structure of (generalized) affine plane, in the sense of incidence geometry.

Tight contact structures and Anosov flows

In this review, we demonstrate how classic and contemporary results on the classification of tight contact structures apply to the problem of existence and uniqueness of Anosov flows on three-manifolds. The ingredients we use are the results of Mitsumatsu on Anosov flows, the homotopy invariant of plane fields as described by Gompf and others, and certain recent classification results of Giroux and Honda. A simple example is a novel proof of the nonexistence of Anosov flows on S 3 using only contact topology (and in particular without use of Novikov's Theorem on foliations).

Entropy rigidity of Anosov flows in dimension three

We show that for a smooth contact Anosov flow on a closed three manifold the measure of maximal entropy is in the Lebesgue class if and only if the flow is up to finite covers conjugate to the geodesic flow of a metric of constant negative curvature on a closed surface.This shows that the ratio between the measure theoretic entropy and the topological entropy of a contact Anosov flow is strictly smaller than one on any closed three manifold which is not a Seifert bundle.

Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows

The paper deals with the billiard flow in the exterior of several strictly convex disjoint domains in the plane with smooth boundaries satisfying an additional (visibility) condition. Using a modification of the technique of Dolgopyat, we get spectral estimates for the Ruelle operator related to a Markov family for the nonwandering (trapping) set of the flow similar to those of Dolgopyat in the case of transitive Anosov flows on compact manifolds with smooth jointly non-integrable horocycle foliations. As a consequence, we get exponential decay of correlation for Hölder continuous potentials on the nonwandering set. Combining the spectral estimate for the Ruelle operator with an argument of Pollicott and Sharp, we also derive the existence of a meromorphic continuation of the dynamical zeta function of the billiard flow to a half-plane Re( s ) < h T - ε, where h T is the topological entropy of the billiard flow, and an asymptotic formula with an error term for the number π(λ) of closed orbits of least period λ > 0.

A short proof of ergodicity of Babillot-Ledrappier measures

Let $M$ be a compact manifold, and let ${\phi _t}$ be a transitive homologically full Anosov flow on $M$. Let $\widetilde {M}$ be a $\mathbb {Z}^d$-cover for $M$, and let $\widetilde {\phi _t}$ be the lift of ${\phi _t}$ to $\widetilde {M}$. Babillot and Ledrappier exhibited a family of measures on $\widetilde {M}$, which are invariant and ergodic with respect to the strong stable foliation of $\widetilde {\phi _t}$. We provide a new short proof of ergodicity.

Error terms for closed orbits of hyperbolic flows

In this paper we obtain a polynomial error term for the closed orbit counting function associated to certain hyperbolic flows. In the case of weak-mixing transitive Anosov flows no further conditions are required; for general weak-mixing hyperbolic flows a diophantine condition on the periods of the closed orbits is required.

Multifractal Analysis of Conformal Axiom A Flows

We develop the multifractal analysis of conformal axiom A flows. This includes the study of the Hausdorff dimension of basic sets of the flow, the description of the dimension spectra for pointwise dimension and for Lyapunov exponents and the multifractal decomposition associated with these spectra. The main tool of study is the thermodynamic formalism for hyperbolic flows by Bowen and Ruelle. Examples include suspensions over axiom A conformal diffeomorphisms, Anosov flows, and in particular, geodesic flows on compact smooth surfaces of negative curvature.

Asymptotic Expansions for Closed Orbits in Homology Classes

In this paper, we study the behaviour of the counting function associated to the closed geodesics lying in a prescribed homology class on a compact negatively curved manifold. Our main result is an asymptotic expansion. We also obtain results in the wider context of periodic orbits of Anosov flows.

Induced maps of hyperbolic Bernoulli systems

Let $(f,T^n,\mu)$ be a linear hyperbolic automorphism of the $n$-torus. We show that if $A\subset T^n$ has a boundary which is a finite union of $C^1$ submanifolds which have no tangents in the stable ($E^s$) or unstable $(E^u)$ direction then the induced map on $A$, $(f_A,A,\mu_A)$ is also Bernoulli. We show that Poincáre maps for uniformly transverse $C^1$ Poincáre sections in smooth Bernoulli Anosov flows preserving a volume measure are Bernoulli if they are also transverse to the strongly stable and strongly unstable foliation.

An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one

As a special case of our results we prove the following. Let A∈ Diff r (M) be an Anosov diffeomorphism. Then there is a C r -neighborhood of A×Id S 1 that contains an open dense set of partially hyperbolic diffeomorphisms that have the accessibility property. If, in addition, A preserves a smooth volume ν and λ is the Lebesgue measure on S 1, then in a neighborhood of A×Id S 1 in Diff ν×λ 2 (M×S 1) there is an open dense set of (stably) ergodic diffeomorphisms. Similar results are true for a neighborhood of the time-1 map of a topologically transitive (respectively volume preserving) Anosov flow. These partially answer a question posed by C. Pugh and M. Shub. We also describe an example of an accessible partially hyperbolic diffeomorphism that is not topologically transitive. This answers a question posed by M. Brin.

Electric fields on a surface of constant negative curvature

A one-parameter family of time reversible Anosov flows is studied; physically, it describes a particle moving on a surface of constant negative curvature under the action of an electric field (corresponding to an automorphic form) and of a ‘thermostatting’ force (given by Gauss's least-constraint principle). We show that the flows are dissipative, in the sense that the average volume contraction rate is positive and the Sinai–Ruelle–Bowen measure is singular with respect to the volume: therefore they verify the assumptions for the validity of the continuous time version of Gallavotti–Cohen's fluctuation theorem for the large fluctuations of the average volume contraction rate. If several independent electric fields are considered, it makes sense to ask for the validity of Onsager's reciprocity: we show, by explicitly computing the relevant transport coefficents, that it is indeed obeyed.

Precise counting results for closed orbits of Anosov flows

We study the problem of counting closed geodesics according to their lengths and under homological constraints on a compact surface of negative curvature. We show how to use Dolgopyat's recent results to obtain a full asymptotic expansion, in addition to the leading term given by Lalley. We first state the properties of the stable and unstable leaves used by Chernov and Dolgopyat; then we introduce the usual transfer operators and we prove the result with the help of a dynamical ζ-function.