On the Equivalence of Equilibrium and Freezing States in Dynamical Systems

Abstract

This paper is concerned with freezing phase transitions in general dynamical systems. A freezing phase transition is one in which, for a given potential $\varphi$, there exists some inverse temperature ${\beta }_0>0$ such that for all $\alpha ,\beta >{\beta }_0$, the collection of equilibrium states for $\alpha \varphi$ and $\beta \varphi$ coincide. In this sense, below the temperature $1/{\beta }_0$, the system "freezes" on a fixed collection of equilibrium states. We show that for a given invariant measure $\mu$, it is no more restrictive that $\mu$ is the freezing state for some potential than it is for $\mu$ to be the equilibrium state for some potential. In fact, our main result applies to any collection of equilibrium states with the same entropy. In the case where the entropy map $h$ is upper semi-continuous, we show any ergodic measure $\mu$ can be obtained as a freezing state for some potential. In this upper semi-continuous setting, we additionally show that the collection of potentials that freeze at a single state is dense in the space of all potentials. However, in the $\mathbb{Z}$ action setting where the dynamical system satisfies specification, the collection of potentials that do not freeze contains a dense $G_{\delta }$.