Multiple phase transitions on compact symbolic systems

Abstract

Let $ϕ:X\to \mathbb R$ be a continuous potential associated with a symbolic dynamical system $T:X\to X$ over a finite alphabet. Introducing a parameter $β>0$ (interpreted as the inverse temperature) we study the regularity of the pressure function $β\mapsto P_{\rm top}(βϕ)$ on an interval $[α,\infty)$ with $α>0$. We say that $ϕ$ has a phase transition at $β_0$ if the pressure function $P_{\rm top}(βϕ)$ is not differentiable at $β_0$. This is equivalent to the condition that the potential $β_0ϕ$ has two (ergodic) equilibrium states with distinct entropies. For any $α>0$ and any increasing sequence of real numbers $(β_n)$ contained in $[α,\infty)$, we construct a potential $ϕ$ whose phase transitions in $[α,\infty)$ occur precisely at the $β_n$'s. In particular, we obtain a potential which has a countably infinite set of phase transitions.