Almost-additive thermodynamic formalism for countable Markov shifts

TL;DR

This paper develops a new thermodynamic formalism for countable Markov shifts using almost-additive sequences, establishing a variational principle and existence of Gibbs measures, with applications to Lyapunov exponents of matrix products.

Contribution

It introduces a novel pressure concept for almost-additive sequences on non-compact shifts, proving a variational principle and existence of equilibrium measures.

Findings

Established a variational principle for the new pressure definition.

Proved existence of Gibbs and equilibrium measures under certain conditions.

Applied the theory to analyze maximal Lyapunov exponents of matrix products.

Abstract

We introduce a definition of pressure for almost-additive sequences of continuous functions defined over (non-compact) countable Markov shifts. The variational principle is proved. Under certain assumptions we prove the existence of Gibbs and equilibrium measures. Applications are given to the study of maximal Lypaunov exponents of product of matrices.