Anisotropic Holder and Sobolev spaces for hyperbolic diffeomorphisms

We study spectral properties of transfer operators for diffeomorphisms T : X -> X on a Riemannian manifold X. Suppose that Omega is an isolated hyperbolic subset for T, with a compact isolating neighborhood V subset of 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.