Volume lemmas for partially hyperbolic endomorphisms and applications

TL;DR

This paper establishes volume lemmas for partially hyperbolic attractors of non-singular endomorphisms, leading to exponential large deviation bounds for Birkhoff averages under certain conditions.

Contribution

It introduces volume lemmas applicable to non-uniformly expanding partially hyperbolic systems and derives large deviation estimates for SRB measures.

Findings

Volume lemmas for Lebesgue and SRB measures on attractors

Exponential large deviation bounds for Birkhoff averages

Applicability to systems with non-uniform cone expansion

Abstract

We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove volume lemmas for both Lebesgue measure on the topological basin of the attractor and the SRB measure supported on the attractor.As a consequence under a mild assumption we prove exponential large deviation bounds for the convergence of Birkhoff averages associated to continuous observables with respect to the SRB measure.