Abstract:
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 arguments are based on the study of the spectral properties of a transfer operator acting on suitable Banach spaces of anisotropic currents.

Abstract:
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.

Abstract:
We consider flows, called $W^{\rm u}$ flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity asumptions, we give a short proof of the strong mixing property of $W^{\rm u}$ flows and we show that $W^{\rm u}$ flows have purely absolutely continuous spectrum in the orthocomplement of the constant functions. As a particular case, we obtain that a class of horocycle flows for compact surfaces of (possibly variable) negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This generalises recent results on time changes of the classical horocycle flows for compact surfaces of constant negative curvature.

Abstract:
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.

Abstract:
In this article, we give a quasi-final classification of quasiconformal Anosov flows. We deduce a very interesting differentable rigidity result for the orbit foliations of hyperbolic manifold of dimension at least three.

Abstract:
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.

Abstract:
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.

Abstract:
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 $\epsilon > 0$. 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 estimates have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, asymptotics of closed orbit counting functions, etc.

Abstract:
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.