Relative equilibrium states and class degree

TL;DR

This paper establishes an upper bound on the number of ergodic measures with maximal entropy and potential in fiber classes of a factor code between shifts, generalizing previous results for the zero potential case.

Contribution

It introduces a bound on ergodic measures maximizing entropy plus potential in fiber classes, extending earlier work to include a general potential function.

Findings

Bound equals class degree when the measure is fully supported.

Generalizes previous zero-potential results.

Provides a new invariant upper bound for ergodic measures.

Abstract

Given a factor code $\pi$ from a shift of finite type $X$ onto a sofic shift $Y$, an ergodic measure $\nu$ on $Y$, and a function $V$ on $X$ with summable variation, we prove an invariant upper bound on the number of ergodic measures on $X$ which project to $\nu$ and maximize $h(\mu) + \int V d\mu$ among all measures in the fiber $\pi^{-1}(\nu)$. If $\nu$ is fully supported, this bound is the class degree of $\pi$. This generalizes a previous result for the special case of $V=0$.