The Parry Simplex: Freezing Phase Transitions for Topological Markov Chains

Abstract

We prove that the pressure function for the Hofbauer potential, defined via the distance to a topological Markov chain (subshift of finite type) X inside the one-sided full shift, exhibits a freezing phase transition. When X is irreducible (in particular when mixing), the post-transition equilibrium measure is unique, equal to the Parry measure of X. When X is reducible with several communicating classes tied for maximal entropy and no chaining among them, the freezing transition persists, but the post-transition equilibrium measure ranges over the full simplex spanned by the Parry measures of the maximal-entropy components.

15 pages