Equilibrium states of the pressure function for products of matrices

TL;DR

This paper investigates the pressure function associated with products of matrices, establishing that for each positive parameter, there are at most d ergodic equilibrium states, each satisfying a Gibbs property.

Contribution

The paper proves a bound on the number of ergodic equilibrium states for the pressure function of matrix products and characterizes their Gibbs properties.

Findings

At most d ergodic q-equilibrium states for each q>0.

Each equilibrium state satisfies a Gibbs property.

The result applies to non-trivial families of complex matrices.

Abstract

Let $\{M_i\}_{i=1}^\ell$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1... i_n\in \{1,..., \ell\}^n$ such that $M_{i_1}... M_{i_n}\neq {\bf 0}$. Let $P \colon (0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^\ell$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property.