Zero temperature limits of equilibrium states for subadditive potentials and approximation of the maximal Lyapunov exponent

TL;DR

This paper investigates the behavior of equilibrium states for subadditive potentials as temperature approaches zero, demonstrating convergence to maximizing measures and approximating maximal Lyapunov exponents via periodic trajectories.

Contribution

It establishes the zero-temperature limits of equilibrium states for subadditive potentials and provides methods to approximate maximal Lyapunov exponents using periodic orbits.

Findings

Equilibrium states converge to maximizing measures as temperature approaches zero.

Lyapunov exponent and entropy of equilibrium states converge to their maximum values.

Maximal Lyapunov exponent can be approximated by periodic trajectories in matrix cocycles.

Abstract

In this paper we study ergodic optimization problems for subadditive sequences of functions on a topological dynamical system. We prove that for $t\rightarrow \infty$ any accumulation point of a family of equilibrium states is a maximizing measure. We show that the Lyapunov exponent and entropy of equilibrium states converge in the limit $t\rightarrow \infty$ to the maximum Lyapunov exponent and entropy of maximizing measures. In the particular case of matrix cocycles we prove that the maximal Lyapunov exponent can be approximated by Lyapunov exponents of periodic trajectories under certain assumptions.