Non-uniform specification and large deviations for weak Gibbs measures

TL;DR

This paper develops bounds for deviation sets in weak Gibbs measures for non-uniformly expanding maps, introducing a weak specification property to handle non-invariant measures and broad classes of dynamical systems.

Contribution

It introduces a measure-theoretical weak form of specification applicable to non-invariant measures and broad classes of non-uniformly expanding maps, extending large deviation principles.

Findings

Bounds for deviation sets in weak Gibbs measures established.

Weak specification property proved for certain non-uniformly expanding maps.

Applicable to quadratic maps and Viana maps.

Abstract

We establish bounds for the measure of deviation sets associated to continuous observables with respect to not necessarily invariant weak Gibbs measures. Under some mild assumptions, we obtain upper and lower bounds for the measure of deviation sets of some non-uniformly expanding maps, including quadratic maps and robust multidimensional non-uniformly expanding local diffeomorphisms. For that purpose, a measure theoretical weak form of specification is introduced and proved to hold for the robust classes of multidimensional nonuniformly expanding local diffeomorphisms and Viana maps.