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:
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 $\Sigma$. 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 homology sphere.

Abstract:
We investigate certain natural connections between subriemannian 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 subriemannian 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 subriemannian geodesic connecting points on the same orbit of the Anosov flow, the energy of the geodesic is shared \emph{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.

Abstract:
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.

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:
We prove strong spectral estimates for Ruelle transfer operators for arbitrary $C^2$ contact Anosov flows in any dimension. For such flows some of the consequences of the main result are: (a) exponential decay of correlations for H\"older continuous observables with respect to any Gibbs measure; (b) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (c) a Prime Orbit Theorem with an exponentially small error. All these results apply for example to geodesic flows on arbitrary compact Riemann manifolds of negative curvature.

Abstract:
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 average of the function D(x)=V(x)-1/2 div(X)/E_u where Eu is the unstable distribution. This band spectrum gives an asymptotic expansion for dynamical correlation functions.

Abstract:
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.