On the Equivalence of Equilibrium and Freezing States in Dynamical Systems

TL;DR

This paper investigates freezing phase transitions in dynamical systems, showing that invariant measures can be characterized as freezing states and that the set of potentials with specific freezing properties is dense under certain conditions.

Contribution

It establishes that invariant measures are as likely to be freezing states as equilibrium states and characterizes the density of potentials with particular freezing behaviors.

Findings

Invariant measures can be realized as freezing states for some potential.

In systems with upper semi-continuous entropy maps, any ergodic measure can be a freezing state.

Potentials that freeze at a single state are dense in the space of all potentials.

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 $\phi$, there exists some inverse temperature $\beta_0 > 0$ such that for all $\alpha, \beta > \beta_0$, the collection of equilibrium states for $\alpha \phi$ and $\beta \phi$ 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 $\Z$ action setting where the dynamical system satisfies specification, the collection of potentials that do not freeze contains a dense $G_\delta$.