Multiple phase transitions on compact symbolic systems

Abstract

Let $\varphi :X\to \mathbb{R}$ be a continuous potential associated with a symbolic dynamical system $T:X\to X$ over a finite alphabet. Introducing a parameter $\beta >0$ (interpreted as the inverse temperature) we study the regularity of the pressure function $\beta \mapsto P_{\mathrm{top}}(\beta \varphi )$ on an interval $[\alpha ,\infty )$ with $\alpha >0$. We say that $\varphi$ has a phase transition at ${\beta }_0$ if the pressure function $P_{\mathrm{top}}(\beta \varphi )$ is not differentiable at ${\beta }_0$. This is equivalent to the condition that the potential ${\beta }_0\varphi$ has two (ergodic) equilibrium states with distinct entropies. For any $\alpha >0$ and any increasing sequence of real numbers $({\beta }_n)$ contained in $[\alpha ,\infty )$, we construct a potential $\varphi$ whose phase transitions in $[\alpha ,\infty )$ occur precisely at the ${\beta }_n$'s. In particular, we obtain a potential which has a countably infinite set of phase transitions.

In this update we present a revised version of the main theorem which now only deals with phase transitions in an interval $[\alpha ,\infty )$ for some fixed $\alpha >0.$