Anosov Flows Research Articles

Nil-Anosov actions

We consider Anosov actions of a Lie group G of dimension k on a closed manifold of dimension \(k+n\). We introduce the notion of Nil-Anosov action of G (which includes the case where G is nilpotent) and establishes the invariance by the entire group G of the associated stable and unstable foliations. We then prove a spectral decomposition Theorem for such an action when the group G is nilpotent. Finally, we focus on the case where G is nilpotent and the unstable bundle has codimension one. We prove that in this case the action is a Nil-extension over an Anosov action of an abelian Lie group. In particular: if \(n \ge 3,\) then the action is topologically transitive, if \(n=2,\) then the action is a Nil-extension over an Anosov flow.

Ergodic components of partially hyperbolic systems

This paper gives a complete classification of the possible ergodic decompositions for certain open families of volume-preserving partially hyperbolic diffeomorphisms. These families include systems with compact center leaves and perturbations of Anosov flows under conditions on the dimensions of the invariant subbundles. The paper further shows that the non-open accessibility classes form a C^1 lamination and gives results about the accessibility classes of non-volume-preserving systems.

Temporal Distributional Limit Theorems for Dynamical Systems

Suppose \(\{T^t\}\) is a Borel flow on a complete separable metric space X, \(f:X\rightarrow \mathbb R\) is Borel, and \(x\in X\). A temporal distributional limit theorem is a scaling limit for the distributions of the random variables \(X_T:=\int _0^t f(T^s x)ds\), where t is chosen randomly uniformly from [0, T], x is fixed, and \(T\rightarrow \infty \). We discuss such laws for irrational rotations, Anosov flows, and horocycle flows.

The semiclassical zeta function for geodesic flows on negatively curved manifolds

We consider the semi-classical (or Gutzwiller-Voros) zeta function for $C^\infty$ contact Anosov flows. Analyzing the spectrum of transfer operators associated to the flow, we prove, for any $\tau>0$, that its zeros are contained in the union of the $\tau$-neighborhood of the imaginary axis, $|\Re(s)|<\tau$, and the region $\Re(s)<-\chi_0+\tau$, up to finitely many exceptions, where $\chi_0>0$ is the hyperbolicity exponent of the flow. Further we show that the zeros in the neighborhood of the imaginary axis satisfy an analogue of the Weyl law.

Sectional Anosov Flows: Existence of Venice Masks with Two Singularities

We show the existence of Venice masks (i.e. nontransitive sectional Anosov flows with dense periodic orbits, Bautista and Morales http://preprint.impa.br/Shadows/SERIE_D/2011/86.html ; Bautista et al. Discr Contin Dyn Syst 19(4):761, 2007; Morales and Pacifico Pac J Math 216(2):327–342, 2004, Morales et al. Pac J Math 229(1):223–232, 2007) containing two equilibria on certain compact 3-manifolds. Indeed, we present two type of examples in which the homoclinic classes composing their maximal invariant set intersect in a very different way.

Partial hyperbolicity and classification: a survey

This paper surveys recent results on classifying partially hyperbolic diffeomorphisms. This includes the construction of branching foliations and leaf conjugacies on three-dimensional manifolds with solvable fundamental group. Classification results in higher-dimensional settings are also discussed. The paper concludes with an overview of the construction of new partially hyperbolic examples derived from Anosov flows.

On the Support of Pollicott–Ruelle Resonanant States for Anosov Flows

We show that all generalized Pollicott-Ruelle resonant states of a topologically transitiv $C^\infty$-Anosov flow with an arbitrary $C^\infty$ potential, have full support.

A Local Trace Formula for Anosov Flows

We prove a local trace formula for Anosov flows. It relates Pollicott–Ruelle resonances to the periods of closed orbits. As an application, we show that the counting function for resonances in a sufficiently wide strip cannot have a sublinear growth. In particular, for any Anosov flow there exist strips with infinitely many resonances.

Explosion of differentiability for equivalencies between Anosov flows on 3-manifolds

For Anosov flows obtained by suspensions of Anosov diffeomorphisms on surfaces, we show the following type of rigidity result: if a topological conjugacy between them is differentiable at a point, then the conjugacy has a smooth extension to the suspended 3 3 -manifold. This result generalizes the similar ones of Sullivan and Ferreira-Pinto for 1-dimensional expanding dynamics and also a result of Ferreira-Pinto for 2-dimensional hyperbolic dynamics.

On the sub-Riemannian geometry of contact Anosov flows

We investigate certain natural connections between sub-Riemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a sub-Riemannian geodesic shared between its projections onto the integrable summands? We show that if the horizontal distribution is the sum of the strong stable and strong unstable distributions of a special type of a contact Anosov flow in three dimensions, then for any short enough sub-Riemannian geodesic connecting points on the same orbit of the Anosov flow, the energy of the geodesic is shared equally between its projections onto the stable and unstable bundles. The proof relies on a connection between the geodesic equations and the harmonic oscillator equation, and its explicit solution by the Jacobi elliptic functions. Using a different idea, we prove an analogous result in higher dimensions for the geodesic flow of a closed Riemannian manifold of constant negative curvature.

Dynamical zeta functions for Anosov flows via microlocal analysis

The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C∞ Anosov flows. More general results have been recently proved by Giulietti–Liverani–Pollicott [GiLiPo] but our approach is different and is based on the study of the generator of the flow as a semiclassical differential operator. The purpose of this article is to provide a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C∞ Anosov flows on compact manifolds: Theorem. Suppose X is a compact manifold and φt : X → X is a C∞ Anosov flow with orientable stable and unstable bundles. Let {γ} denote the set of primitive orbits of φt, with T ] γ their periods. Then the Ruelle zeta function, ζR(λ) = ∏

Anomalous partially hyperbolic diffeomorphisms I: dynamically coherent examples

We build an example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism f on a closed 3-manifold with exponential growth in its fundamental group such that f n is not isotopic to the identity for all n 0. This example contradicts a conjecture in [HHU1]. The main idea is to consider a well-understood time-t map of a non-transitive Anosov flow and then carefully compose with a Dehn twist.

Invariant distributions and X-ray transform for Anosov flows

For Anosov flows preserving a smooth measure on a closed manifold $\mathcal{M}$, we define a natural self-adjoint operator $\Pi$ which maps into the space of invariant distributions in $\cap_{u<0} H^{u}(\mathcal{M})$ and whose kernel is made of coboundaries in $\cup_{s>0} H^{s}(\mathcal{M})$. We describe relations to Livsic theorem and recover regularity properties of cohomological equations using this operator. For Anosov geodesic flows on the unit tangent bundle $\mathcal{M}=SM$ of a compact manifold, we apply this theory to study questions related to $X$-ray transform on symmetric tensors on $M$: in particular we prove that injectivity implies surjectivity of X-ray transform, and we show injectivity for surfaces.

A spectral-like decomposition for transitive Anosov flows in dimension three

Given a (transitive or non-transitive) Anosov vector field X on a closed three dimensional manifold M, one may try to decompose (M, X) by cutting M along tori and Klein bottles transverse to X. We prove that one can find a finite collection $$\{S_1,\dots ,S_n\}$$ of pairwise disjoint, pairwise non-parallel tori and Klein bottles transverse to X, such that the maximal invariant sets $$\Lambda _1,\dots ,\Lambda _m$$ of the connected components $$V_1,\dots ,V_m$$ of $$M-(S_1\cup \dots \cup S_n)$$ satisfy the following properties: To a certain extent, the sets $$\Lambda _1,\dots ,\Lambda _m$$ are analogs (for Anosov vector field in dimension 3) of the basic pieces which appear in the spectral decomposition of a non-transitive axiom A vector field. Then we discuss the uniqueness of such a decomposition: we prove that the pieces of the decomposition $$V_1,\dots ,V_m$$ , equipped with the restriction of the Anosov vector field X, are “almost unique up to topological equivalence”.

On expanding foliations

Certain families of manifolds which support Anosov flows do not support foliations which both are expanding under a dynamical system and have quasi-isometrically embedded leaves.

Livšic measurable rigidity for 𝒞¹ generic volume-preserving Anosov systems

In this paper, we prove that for C 1 \mathcal {C}^1 generic volume-preserving Anosov diffeomorphisms of a compact connected Riemannian manifold, the Livšic measurable rigidity theorem holds. We also give a parallel result for C 1 \mathcal {C}^1 generic volume-preserving Anosov flows.

Rigidity of Hamenstädt metrics of Anosov flows

Let $\varphi$ be a $C^\infty$ transversely symplectic topologically mixing Anosov flow such that $dim E^{su}\geq 2$. We suppose that the weak distributions of $\varphi$ are $C^1$. If the length Hamenstadt metrics of $\varphi$ are sub-Riemannian then we prove that the weak distributions of $\varphi$ are necessarily $C^\infty$. Combined with our previous rigidity result in [5] we deduce the classification of such Anosov flows with $C^1$ weak distributions provided that the length Hamenstadt metrics are sub-Riemannian.

Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group

A classification of partially hyperbolic diffeomorphisms on 3-dimensional manifolds with (virtually) solvable fundamental group is obtained. If such a diffeomorphism does not admit a periodic attracting or repelling two-dimensional torus, it is dynamically coherent and leaf conjugate to a known algebraic example. This classification includes manifolds which support Anosov flows, and it confirms conjectures by Rodriguez Hertz--Rodriguez Hertz--Ures and Pujals in the specific case of solvable fundamental group.

Power spectrum of the geodesic flow on hyperbolic manifolds

We describe the complex poles of the power spectrum of correlations for the geodesic flow on compact hyperbolic manifolds in terms of eigenvalues of the Laplacian acting on certain natural tensor bundles. These poles are a special case of Pollicott-Ruelle resonances, which can be defined for general Anosov flows. In our case, resonances are stratified into bands by decay rates. The proof also gives an explicit relation between resonant states and eigenstates of the Laplacian.

Global cross sections for Anosov flows

We provide a new criterion for the existence of a global cross section to a volume-preserving Anosov flow. The criterion is expressed in terms of expansion and contraction rates of the flow and generalizes known results of this type.