Relative equilibrium states and class degree

Abstract

Given a factor code $\unicode[STIX]{x1D70B}$ from a shift of finite type $X$ onto a sofic shift $Y$ , an ergodic measure $\unicode[STIX]{x1D708}$ on $Y$ , and a function $V$ on $X$ with sufficient regularity, we prove an invariant upper bound on the number of ergodic measures on $X$ which project to $\unicode[STIX]{x1D708}$ and maximize the measure pressure $h(\unicode[STIX]{x1D707})+\int V\,d\unicode[STIX]{x1D707}$ among all measures in the fiber $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D708})$ . If $\unicode[STIX]{x1D708}$ is fully supported, this bound is the class degree of $\unicode[STIX]{x1D70B}$ . This generalizes a previous result for the special case of $V=0$ and thus settles a conjecture raised by Allahbakhshi and Quas.