Multifractal analysis of the irregular set for almost-additive sequences via large deviations

TL;DR

This paper develops a multifractal analysis framework for almost-additive sequences using large deviations, providing new estimates for topological pressure and applications to dynamical systems and ergodic theory.

Contribution

It introduces a novel approach to analyze irregular sets for almost-additive sequences via free energy and large deviations, extending existing multifractal analysis methods.

Findings

Derived estimates for topological pressure of irregular sets

Applied results to Lyapunov exponents in dynamical systems

Extended Shannon-McMillan-Breiman theorem for Gibbs measures

Abstract

In this paper we introduce a notion of free energy and large deviations rate function for asymptotically additive sequences of potentials via an approximation method by families of continuous potentials. We provide estimates for the topological pressure of the set of points whose non-additive sequences are far from the limit described through Kingman's sub-additive ergodic theorem and give some applications in the context of Lyapunov exponents for diffeomorphisms and cocycles, and Shannon-McMillan-Breiman theorem for Gibbs measures.