Anosov Flows Research Articles

The Fried conjecture in small dimensions

We study the twisted Ruelle zeta function $\zeta_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $\zeta_X(0)$. In higher dimensions, we show more generally that $\zeta_X(0)$ is locally constant with respect to the vector field $X$ under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic $3$-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flows in variable negative curvature where Fried conjecture holds true.

On the Relation between Enhanced Dissipation Timescales and Mixing Rates

AbstractWe study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect that causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a timescale faster than the purely diffusive one. We establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation timescales. The proofs are based on a contradiction argument that takes advantage of the cascading mechanism due to mixing, an estimate of the distance between the inviscid and viscous dynamics, and an optimization step in the frequency cutoff.Thanks to the generality and robustness of our approach, we are able to apply our abstract results to a number of problems. For instance, we prove that contact Anosov flows obey logarithmically fast dissipation timescales. To the best of our knowledge, this is the first example of a flow that induces an enhanced dissipation timescale faster than polynomial. Other applications include passive scalar evolution in both planar and radial settings and fractional diffusion. © 2019 Wiley Periodicals, Inc.

Periodic measures and partially hyperbolic homoclinic classes

In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to the global setting of partially hyperbolic diffeomorphisms with one-dimensional center. When both strong stable and unstable foliations are minimal, we get that the closure of the set of ergodic measures is the union of two convex sets corresponding to the two possible $s$-indices; these two convex sets intersect along the closure of the set of non-hyperbolic ergodic measures. That is the case for robustly transitive perturbations of a time-one map of a transitive Anosov flow, or of the skew product of an Anosov torus diffeomorphism by a rotation of the circle.

Topological entropy on points without physical-like behaviour

We study a class of asymptotically entropy-expansive $C^1$ diffeomorphisms with dominated splitting on a compact manifold $M$, that satisfy the specification property. This class includes, in particular, transitive Anosov diffeomorphisms and time-one maps of transitive Anosov flows. We consider the nonempty set of physical-like measures that attracts the empirical probabilities (i.e. the time averages) of Lebesgue-almost all the orbits. We define the set $I_f \cap \Gamma_f \subset M$ of irregular points without physical-like behaviour. We prove that, if not all the invariant measures of $f$ satisfy Pesin Entropy Formula (for instance in the Anosov case), then $I_f \cap \Gamma_f$ has full topological entropy. We also obtain this result for some class of asymptotically entropy-expansive continuous maps on a compact metric space, if the set of physical-like measures are equilibrium states with respect to some continuous potential. Finally, we prove that also the set $(M \setminus I_f) \cap \Gamma_f$ of regular points without physical-like behaviour, has full topological entropy.

Information geometry in a global setting

We begin a global study of information geometry. In this article, we describe the geometry of normal distributions by means of positive and negative contact structures associated to the suspension Anosov flows on $Sol^3$-manifolds.

Engel structures and weakly hyperbolic flows on four-manifolds

We study pairs of Engel structures on four-manifolds whose intersection has constant rank one and which define the same even contact structure, but induce different orientations on it. We establish a correspondence between such pairs of Engel structures and a class of weakly hyperbolic flows. This correspondence is analogous to the correspondence between bi-contact structures and projectively or conformally Anosov flows on three-manifolds found by Eliashberg–Thurston and by Mitsumatsu.

Holomorphic differentials, thermostats and Anosov flows

We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian two-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties. In particular, we prove that they admit a dominated splitting and we identify special cases in which the flows are Anosov. In the latter case, we study when they admit an invariant measure in the Lebesgue class and the regularity of the weak foliations.

A note on Anosov flows of non-compact Riemannian manifolds

In this note we formulate a condition for complete non-compact Riemannian manifolds, which implies no conjugate points in case that the geodesic flow is Anosov with respect to the Sasaki metric.

Exponential mixing for generic volume-preserving Anosov flows in dimension three

Let $M$ be a closed 3-dimensional Riemann manifold and let $3\le r\le \infty$. We prove that there exists an open dense subset in the space of $C^r$ volume-preserving Anosov flows on $M$ such that all the flows in it are exponentially mixing.

Uniformly quasiconformal partially hyperbolic systems

We study smooth volume-preserving perturbations of the time-1 map of the geodesic flow $\psi_{t}$ of a closed Riemannian manifold of dimension at least three with constant negative curvature. We show that such a perturbation has equal extremal Lyapunov exponents with respect to volume within both the stable and unstable bundles if and only if it embeds as the time-1 map of a smooth volume-preserving flow that is smoothly orbit equivalent to $\psi_{t}$. Our techniques apply more generally to give an essentially complete classification of smooth, volume-preserving, dynamically coherent partially hyperbolic diffeomorphisms which satisfy a uniform quasiconformality condition on their stable and unstable bundles and have either compact center foliation with trivial holonomy or are obtained as perturbations of the time-1 map of an Anosov flow.

Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms

We consider $ \mathit{Sp}\left( 2\mathit{d},\mathbb{R} \right)$ cocycles over two classes of partially hyperbolic diffeomorphisms: having compact center leaves and time one maps of Anosov flows. We prove that the Lyapunov exponents are non-zero in an open and dense set in the Hölder topology.

Seifert manifolds admitting partially hyperbolic diffeomorphisms

We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow.

Horocyclic invariance of Ruelle resonant states for contact Anosov flows in dimension $3$

Horocyclic invariance of Ruelle resonant states for contact Anosov flows in dimension $3$

Equidistribution of holonomy in homology classes for Anosov flows

We obtain equidistribution results for the holonomies of periodic orbits of Anosov flows lying in a prescribed homology class. We apply this to frame flows.

Counting periodic orbits of Anosov flows in free homotopy classes

The main result of this article is that if a 3-manifold M supports an Anosov flow, then the number of conjugacy classes in the fundamental group of M grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show that, when the flow is transitive, the exponential growth rate is exactly the topological entropy of the flow. We also show that taking only the shortest orbit representatives in each conjugacy classes still yields Bowen’s version of the measure of maximal entropy. These results are achieved by obtaining counting results on the growth rate of the number of periodic orbits inside a free homotopy class . In the first part of the article, we also construct many examples of Anosov flows having some finite and some infinite free homotopy classes of periodic orbits, and we also give a characterization of algebraic Anosov flows as the only \mathbb R -covered Anosov flows up to orbit equivalence and finite lifts that do not admit at least one infinite free homotopy class of periodic orbits.

On the topology of the Lorenz system.

We present a new paradigm for three-dimensional chaos, and specifically for the Lorenz equations. The main difficulty in these equations and for a generic flow in dimension 3 is the existence of singularities. We show how to use knot theory as a way to remove the singularities. Specifically, we claim: (i)for certain parameters, the Lorenz system has an invariant one-dimensional curve, which is a trefoil knot. The knot is a union of invariant manifolds of the singular points. (ii)The flow is topologically equivalent to an Anosov flow on the complement of this curve, and moreover to a geodesic flow. (iii)When varying the parameters, the system exhibits topological phase transitions, i.e. for special parameter values, it will be topologically equivalent to an Anosov flow on a knot complement. Different knots appear for different parameter values and each knot controls the dynamics at nearby parameters.

Transverse foliations on the torus $\mathbb T^2$ and partially hyperbolic diffeomorphisms on 3-manifolds

In this paper, we prove that given two C^1 foliations \mathcal{F} and \mathcal{G} on \mathbb{T}^2 which are transverse, there exists a non-null homotopic loop {\{\Phi_t\}_{t\in[0,1]}} in \mathrm {Diff}^{1}(\mathbb T^2) such that {\Phi_t(\mathcal{F})\pitchfork \mathcal{G}} for every t\in[0,1] , and \Phi_0=\Phi_1= \mathrm {Id} . As a direct consequence, we get a general process for building new partially hyperbolic diffeomorphisms on closed 3 -manifolds. [4] built a new example of dynamically coherent non-transitive partially hyperbolic diffeomorphism on a closed 3 -manifold; the example in [4] is obtained by composing the time t map, t>0 large enough, of a very specific non-transitive Anosov flow by a Dehn twist along a transverse torus. Our result shows that the same construction holds starting with any non-transitive Anosov flow on an oriented 3-manifold. Moreover, for a given transverse torus, our result explains which type of Dehn twists lead to partially hyperbolic diffeomorphisms.

Quantitative Pesin theory for Anosov diffeomorphisms and flows

Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure-preserving dynamical system is explicitly controlled. In uniformly hyperbolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows and consider counterexamples illustrating the necessity of suitable conditions on the cocycle.

Building Anosov flows on 3–manifolds

We prove a result allowing to build (transitive or non-transitive) Anosov flows on 3-manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications; for example: 1. we build a 3-manifold supporting both of a transitive Anosov vector field and a non-transitive Anosov vector field; 2. for any n, we build a 3-manifold M supporting at least n pairwise different Anosov vector fields; 3. we build transitive attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive attractors; 4. we build a transitive Anosov vector field admitting infinitely many pairwise non-isotopic trans- verse tori.

Rigidity for partially hyperbolic diffeomorphisms

In this work we completely classify$C^{\infty }$conjugacy for smooth conservative (pointwise) partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain$C^{\infty }$conjugacy to its linear Anosov automorphism. Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows.J. Eur. Math. Soc. (JEMS)17(6) (2015), 1435–1462] proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute version of absolute continuity is the appropriate scenario for our context since it is not possible to obtain a result analogous to that of Avila, Viana and Wilkinson for our class of maps, for absolute continuity alone fails miserably to imply smooth rigidity for our class of maps. Our theorem is a global rigidity result as we do not assume the diffeomorphism to be at some distance from the linear Anosov automorphism. We also do not assume ergodicity. In particular, a metric condition on the center foliation implies ergodicity and$C^{\infty }$center foliation.