Large deviations for systems with non-uniform structure

TL;DR

This paper develops large deviations principles for symbolic systems with non-uniform structures using weak Gibbs and specification properties, applicable to various shifts and factors.

Contribution

It introduces a novel approach employing weak Gibbs and horseshoe theorems to establish large deviations for non-uniform symbolic systems.

Findings

Large deviations principles derived for a broad class of symbolic systems.

Application to $eta$-shifts, $S$-gap shifts, and their factors.

Established a horseshoe theorem for these systems.

Abstract

We use a weak Gibbs property and a weak form of specification to derive level-2 large deviations principles for symbolic systems equipped with a large class of reference measures. This has applications to a broad class of symbolic systems, including $\beta$-shifts, $S$-gap shifts, and their factors. A crucial step in our approach is to prove a `horseshoe theorem' for these systems.