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Let $\phi:X\to \mathbb R$ be a continuous potential associated with asymbolic dynamical system $T:X\to X$ over a finite alphabet. Introducing aparameter $\beta>0$ (interpreted as the inverse temperature) we study theregularity of the pressure function $\beta\mapsto P_{\rm top}(\beta\phi)$ on aninterval $[\alpha,\infty)$ with $\alpha>0$. We say that $\phi$ has a phasetransition at $\beta_0$ if the pressure function $P_{\rm top}(\beta\phi)$ isnot differentiable at $\beta_0$. This is equivalent to the condition that thepotential $\beta_0\phi$ has two (ergodic) equilibrium states with distinctentropies. 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 countablyinfinite set of phase transitions.

Multiple phase transitions on compact symbolic systems