Good Banach spaces for piecewise hyperbolic maps via interpolation

TL;DR

This paper establishes a new framework for analyzing piecewise hyperbolic maps using anisotropic Sobolev spaces, leading to spectral gap results and existence of physical measures, unifying various previous approaches.

Contribution

Introduces a weak transversality condition and applies complex interpolation to obtain spectral bounds for transfer operators of hyperbolic maps.

Findings

Spectral gap for transfer operators under new conditions

Existence of finitely many physical measures with full basin

Unifies analysis of hyperbolic and expanding maps

Abstract

We introduce a weak transversality condition for piecewise C^{1+\alpha} and piecewise hyperbolic maps which admit a C^{1+\alpha} stable distribution. We show good bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel-Lizorkin type. In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) and applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results.