We show, using semiclassical measures and unstable derivatives, that a smooth vector field $X$ generating a contact Anosov flow on a $3$-dimensional manifold $\mathcal{M}$ has only finitely many Ruelle resonances in the vertical strips $\{ s\in \mathbb{C}\ |\ {\rm Re}(s)\in [-ν_{\min}+ε,-\frac{1}{2}ν_{\max}-ε]\cup [-\frac{1}{2}ν_{\min}+ε,0]\}$ for all $ε>0$, where $0<ν_{\min}\leq ν_{\max}$ are the minimal and maximal expansion rates of the flow (the first strip only makes sense if $ν_{\min}>ν_{\max}/2$). We also show polynomial bounds in $s$ for the resolvent $(-X-s)^{-1}$ as $|{\rm Im}(s)|\to \infty$ in Sobolev spaces, and obtain similar results for cases with a potential. This is a short proof of a particular case of the results by Faure-Tsujii in \cite{FaTs1,FaTs2,FaTs3}, using that $\dim E_u=\dim E_s=1$.
We show that for smooth contact Anosov flows in dimension 3, the resonant states associated to the first band of Ruelle resonances are distributions that are killed by the unstable derivative.
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.
I show that the dynamical determinant, associated to an Anosov diffeomorphism, is the Fredholm determinant of the corresponding Ruelle-Perron-Frobenius transfer operator acting on appropriate Banach spaces. As a consequence it follows, for example, that the zeroes of the dynamical determinant describe the eigenvalues of the transfer operator and the Ruelle resonances and that, for $\Co^\infty$ Anosov diffeomorphisms, the dynamical determinant is an entire function.
Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (-i)V, called Ruelle resonances, close to the real axis and for large real parts.
We prove exponential decay of correlations for Hölder continuous observables with respect to any Gibbs measure for contact Anosov flows admitting Pesin sets with exponentially small tails. This is achieved by establishing strong spectral estimates for certain Ruelle transfer operators for such flows.
The Ruelle resonances of a dynamical system are spectral data describing the precise asymptotics of correlations. We classify them completely for a class of chaotic two-dimensional maps, the linear pseudo-Anosov maps, in terms of the action of the map on cohomology. As applications, we obtain a full description of the distributions which are invariant under the linear flow in the stable direction of such a linear pseudo-Anosov map, and we solve the cohomological equation for this flow.
Combining microlocal methods and a cohomological theory developped by J. Taylor, we define for $\mathbb{R}^κ$-Anosov actions a notion of joint Ruelle resonance spectrum. We prove that these Ruelle-Taylor resonances fit into a Fredholm theory, are intrinsic and form a discrete subset of $\mathbb{C}^κ$, with $λ=0$ being always a leading resonance. The joint resonant states at $0$ give rise to some new measures of SRB type and the mixing properties of these measures are related to the existence of purely imaginary resonances. The spectral theory developed in this article applies in particular to …
We consider Anosov flows on closed 3-manifolds preserving a volume form $Ω$. Following Dyatlov and Zworski (2017) we study spaces of invariant distributions with values in the bundle of exterior forms whose wavefront set is contained in the dual of the unstable bundle. Our first result computes the dimension of these spaces in terms of the first Betti number of the manifold, the cohomology class $[ι_{X}Ω]$ (where $X$ is the infinitesimal generator of the flow) and the helicity. These dimensions coincide with the Pollicott-Ruelle resonance multiplicities under the assumption of $\textit{semisim…
We give a sharp polynomial bound on the number of Pollicott-Ruelle resonances. These resonances, which are complex numbers in the lower half-plane, appear in expansions of correlations for Anosov contact flows. The bounds follow the tradition of upper bounds on the number of scattering resonances and improve a recent bound of Faure-Sjöstrand. The complex scaling method used in scattering theory is replaced by an approach using exponentially weighted spaces introduced by Helffer-Sjöstrand in scattering theory and by Faure-Sjöstrand in the theory of Anosov flows.
For Anosov flows on compact Riemann manifolds we study the rate of decay along the flow of diameters of balls $B^s(x,\ep)$ on local stable manifolds at Lyapunov regular points $x$. We prove that this decay rate is similar for all sufficiently small values of $ε> 0$. From this and the main result in \cite{kn:St1}, we derive strong spectral estimates for Ruelle transfer operators for contact Anosov flows with Lipschitz local stable holonomy maps. These apply in particular to geodesic flows on compact locally symmetric manifolds of strictly negative curvature. As is now well known, such spectral …
If X is a contact Anosov vector field on a smooth compact manifold M and V is a smooth function on M, it is known that the differential operator A=-X+V has some discrete spectrum called Ruelle-Pollicott resonances in specific Sobolev spaces. We show that for |Im(z)| large the eigenvalues of A are restricted to vertical bands and in the gaps between the bands, the resolvent of A is bounded uniformly with respect to |Im(z)|. In each isolated band the density of eigenvalues is given by the Weyl law. In the first band, most of the eigenvalues concentrate of the vertical line Re(z)=< D >, the space…
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.
We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension $d=2$ we show that the transfer operator associated to …
In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact Anosov flows. The ultimate aim is to establish exponential decay of correlations for Hölder observables with respect to a very general class of Gibbs measures. The approach invented in 1997 by Dolgopyat \cite{D1} and further developed in \cite{St2} is substantially refined here, allowing to deal with much more general situations than before, although we still restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which pro…
The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C^\infty Anosov flows. More general results have been recently proved by Giulietti-Liverani-Pollicott [arXiv:1203.0904] but our approach is different and is based on the study of the generator on the flow as a semiclassical differential operator.
For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to these of Dolgopyat (1998) for geodesic flows on compact surfaces (for general potentials)and transitive Anosov flows on compact manifolds with C^1 jointly non-integrable horocycle foliations (for the Sinai-Bowen-Ruelle potential). Here we deal with general potentials. As is now well known, such results have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, clo…
These notes are based on lectures given by the author at the Summer School on Teichmüller dynamics, mapping class groups and applications in Grenoble, France, in June 2018 and at the Oberwolfach Seminar on Anisotropic Spaces and their Applications to Hyperbolic and Parabolic Systems in June 2019. We derive results about the so-called Ruelle resonances and the asymptotics of correlations for several classes of systems from known results on cohomological equations and invariant distributions for the respective unstable vector fields. In particular, we consider pseudo-Anosov diffeomorphisms on su…
This errata fixes a mistake in the part of Giulietti, P.; Liverani, C.; Pollicott, M. Anosov flows and dynamical zeta functions. Ann. of Math. (2) {\bf 178} (2013), no. 2, 687--773, which proves a spectral gap for contact Anosov flows with respect to the measure of maximal entropy (Section 7). However, the first part of the paper, in which it is proved that the Ruelle zeta function is meromorphic, is unaffected.
We study real analytic perturbations of hyperbolic linear automorphisms on the 2-torus. The Koopman and the transfer operator are nuclear of order 0 when acting on a suitable Hilbert space. We show the generic existence of non-trivial Ruelle resonances for both operators. We prove that some of the perturbations preserve the volume and some of them do not.
Exponential decay of correlations for $\Co^{(4)}$ Contact Anosov flows is established. This implies, in particular, exponential decay of correlations for all smooth geodesic flows in strictly negative curvature.
We study the Ruelle and Selberg zeta functions for $\Cs^r$ Anosov flows, $r > 2$, on a compact smooth manifold. We prove several results, the most remarkable being: (a) for $\Cs^\infty$ flows the zeta function is meromorphic on the entire complex plane; (b) for contact flows satisfying a bunching condition (e.g. geodesic flows on manifolds of negative curvature better than $\frac 19$-pinched) the zeta function has a pole at the topological entropy and is analytic in a strip to its left; (c) under the same hypotheses as in (b) we obtain sharp results on the number of periodic orbits. Our argume…
The first half of this paper is concerned with the topology of the space $\AAA(M)$ of (not necessarily contact) Anosov vector fields on the unit tangent bundle $M$ of closed oriented hyperbolic surfaces $Σ$. We show that there are countably infinite connected components of $\AAA(M)$, each of which is not simply connected. In the second part, we study contact Anosov flows. We show in particular that the time changes of contact Anosov flows form a $C^1$-open subset of the space of the Anosov flows which leave a particular $C^\infty$ volume form invariant, if the ambiant manifold is a rational ho…
We show that the differential in positive equivariant symplectic homology or linearized contact homology vanishes for non-degenerate Reeb flows with a continuous invariant Lagrangian subbundle (e.g. Anosov Reeb flows). Several applications are given, including obstructions to the existence of these flows and abundance of periodic orbits for contact forms representing an Anosov contact structure.
We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show $\mathbb{S}^3$ does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a contact structure implying that a Reeb field is Anosov. From this we can give curvature conditions on a metric compatible with a contact structure that implies universal tightness of the contact structure among other things.
We study the cohomological pressure introduced by R.Sharp (defined by using topological pressures of certain potentials of Anosov flows). In particular, we get the rigidity in the case that this pressure coincides with the metrical entropy, generalising related rigidity results of A.Katok and P. Foulon.
We discuss a metric description of the divergence of a (projectively) Anosov flow in dimension 3, in terms of its associated growth rates and give metric and contact geometric characterizations of when a projectively Anosov flow is Anosov. Then, we study the symmetries that the existence of an invariant volume form yields on the geometry of an Anosov flow, from various viewpoints of the theory of contact hyperbolas, Reeb dynamics and Liouville geometry, and give characterizations of when an Anosov flow is volume preserving in terms of those theories. We finally use our study to show that the b…
We study the twisted Ruelle zeta function $ζ_X(s)$ 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 $ζ_X(0)$. In higher dimensions, we show more generally that $ζ_X(0)$ 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 flo…
Let $X \subset \mathbb{R}^4$ be a convex domain with smooth boundary $Y$. We use a relation between the extrinsic curvature of $Y$ and the Ruelle invariant $\text{Ru}(Y)$ of the natural Reeb flow on $Y$ to prove that there exist constants $C > c > 0$ independent of $Y$ such that \[c < \frac{\text{Ru}(Y)^2}{\text{vol}(X)} \cdot \text{sys}(Y) < C\] Here $\text{sys}(Y)$ is the systolic ratio, i.e. the square of the minimal period of a closed Reeb orbit of $Y$ divided by twice the volume of $X$. We then construct dynamically convex contact forms on $S^3$ that violate this bound using methods of Ab…
This paper is about spectral properties of transfer operators for contact Anosov flows. The main result gives the essential spectral radius of the transfer operators acting on the so-called anisotropic Sobolev space exactly in terms of dynamical exponents. Also we provide a simplified proof by using the FBI transform.
For any $C^r$ contact Anosov flow with $r\ge 3$, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all $C^r$ functions, such that the transfer operators for the flow extend to them boundedly and that the extensions are quasi-compact. Further we give explicit bounds on the essential spectral radii of the extensions in terms of the differentiability r and the hyperbolicity exponents of the flow.
For the time-one map $f$ of a contact Anosov flow on a compact Riemann manifold $M$, satisfying a certain regularity condition, we show that given a Gibbs measure on $M$, a sufficiently large Pesin regular set $P_0$ and an arbitrary $δ\in (0,1)$, there exist positive constants $C$ and $c$ such that for any integer $n \geq 1$, the measure of the set of those $x\in M$ with $f^k(x) \notin P_0$ for at least $δn$ values of $k = 0,1, \ldots,n-1$ does not exceed $C e^{-cn}$.
On a compact manifold of any dimension $d\geq 3$, we show that joint non-integrability of the stable and unstable foliation of a hyperbolic attractor with one-dimensional expanding direction, for a vector field of class $C^2$, implies exponential mixing with respect to its physical measure. Consequently, the set of Axiom A vector fields which mix exponentially with respect to the physical measure of its non-trivial attractors contains a $C^1$-open and $C^2$-dense subset of the set of all Axiom A vector fields. Moreover, for codimension one $C^2$ Anosov flows in any dimension $d\geq 3$, if the …
