Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms

Abstract

We study spectral properties of transfer operators for diffeomorphisms T:X→X on a Riemannian manifold X. Suppose that Ω is an isolated hyperbolic subset for T, with a compact isolating neighborhood V⊂X. We first introduce Banach spaces of distributions supported on V, which are anisotropic versions of the usual space of C p functions C p (V) and of the generalized Sobolev spaces W p,t (V), respectively. We then show that the transfer operators associated to T and a smooth weight g extend boundedly to these spaces, and we give bounds on the essential spectral radii of such extensions in terms of hyperbolicity exponents.