On the Abundance of Phase Transitions in Dynamical Systems

In this work, we investigate the mechanisms that trigger phase transitions in both discrete-time and continuous-time dynamical systems. We prove a simple topological condition that implies the existence of Hölder continuous potentials displaying phase transitions. As a consequence, we show that phase transitions are typical in several scenarios. Specifically, for $C^1$ diffeomorphisms and $C^1$ vector fields, we show that phase transitions are generic among non-transitive systems. In low dimensions, we conclude that they are generic for non-Anosov systems. Finally, in the $C^0$ setting, we prove that on any compact topological manifold of dimension different from 4, there exists a dense set of homeomorphisms with finite entropy that display phase transitions.