Multiple phase transitions on compact symbolic systems

Tamara Kucherenko, Antony Quas, Christian Wolf

Published 2020-06-24Version 1

Let $\phi:X\to R$ be a continuous potential associated with a symbolic dynamical system $T:X\to X$ over a finite alphabet. Introducing a parameter $\beta>0$ (interpreted as the inverse temperature) we study the regularity of the pressure function $\beta\mapsto P_{\rm top}(\beta\phi)$. We say that $\phi$ has a phase transition at $\beta_0$ if the pressure function $P_{\rm top}(\beta\phi)$ is not differentiable at $\beta_0$. This is equivalent to the condition that the potential $\beta_0\phi$ has two (ergodic) equilibrium states with distinct entropies. In this paper we construct for any given countable set of positive real numbers $\{\beta_n\}$ a potential $\phi$ which has phase transitions precisely at $\beta_n$. Taking $\{\beta_n\}$ to be finite, we see that a continuous potential can have any finite number of phase transitions occurring at any set of predetermined points. Taking $\{\beta_n\}$ to be infinite, we obtain a potential whose set of phase transitions is countably infinite.