A necessary and sufficient condition for a matrix equilibrium state to be mixing

TL;DR

This paper establishes a precise algebraic criterion that determines when matrix equilibrium states are mixing, completing previous work that only provided sufficient conditions.

Contribution

It provides a necessary and sufficient condition for mixing of matrix equilibrium states, advancing the understanding of their ergodic properties.

Findings

Complete characterization of mixing conditions for matrix equilibrium states.

Clarification of algebraic properties related to ergodic behavior.

Extension of previous results on entropy and Bernoulli properties.

Abstract

Since the 1970s there has been a rich theory of equilibrium states over shift spaces associated to H\"older-continuous real-valued potentials. The construction of equilibrium states associated to matrix-valued potentials is much more recent, with a complete description of such equilibrium states being achieved in 2011 by D.-J. Feng and A. K\"aenm\"aki. In a recent article the author investigated the ergodic-theoretic properties of these matrix equilibrium states, attempting in particular to give necessary and sufficient conditions for mixing, positive entropy, and the property of being a Bernoulli measure with respect to the natural partition, in terms of the algebraic properties of the semigroup generated by the matrices. Necessary and sufficient conditions were successfully established for the latter two properties but only a sufficient condition for mixing was given. The purpose of this note is to complete that investigation by giving a necessary and sufficient condition for a matrix equilibrium state to be mixing.