Multiple phase transitions on compact symbolic systems

Let $phi:Xto mathbb R$ be a continuous potential associated with a symbolic dynamical system $T:Xto X$ over a finite alphabet. Introducing a parameter $beta>0$ (interpreted as the inverse temperature) we study the regularity of the pressure function $betamapsto P_{rm top}(betaphi)$ on an interval $[alpha,infty)$ with $alpha>0$. We say that $phi$ has a phase transition at $beta_0$ if the pressure function $P_{rm top}(betaphi)$ is not differentiable at $beta_0$. This is equivalent to the condition that the potential $beta_0phi$ 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 $phi$ 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.