  Let $\phi:X\to \mathbb R$ 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 $\beta\mapsto P_{\rm top}(\beta\phi)$ on an
interval $[\alpha,\infty)$ with $\alpha>0$. We say that $\phi$ has a phase
transition at $\beta_0$ if the pressure function $P_{\rm top}(\beta\phi)$ 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. 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.
