Multiple phase transitions on compact symbolic systems

Abstract

Let ϕ:X→R be a continuous potential associated with a symbolic dynamical system T:X→X over a finite alphabet. Introducing a parameter β>0 (interpreted as the inverse temperature) we study the regularity of the pressure function β↦Ptop(βϕ) on an interval [α,∞) with α>0. We say that ϕ has a phase transition at β0 if the pressure function Ptop(βϕ) is not differentiable at β0. This is equivalent to the condition that the potential β0ϕ has two (ergodic) equilibrium states with distinct entropies. For any α>0 and any increasing sequence of real numbers (βn) contained in [α,∞), we construct a potential ϕ whose phase transitions in [α,∞) occur precisely at the βn's. In particular, we obtain a potential which has a countably infinite set of phase transitions.