On the Equivalence of Equilibrium and Freezing States in Dynamical Systems
C. Evans Hedges
TL;DR
The paper develops a general framework for freezing phase transitions in dynamical systems by linking freezing states to equilibrium states under a fixed potential. It proves that a nonempty set of invariant measures can be realized as the freezing set for some ψ if and only if entropy is constant on that set and the set is an equilibrium set for some φ, with corollaries showing that any ergodic measure is a freezing state for some potential under upper semicontinuity of the entropy map. It further establishes that freezing-favorable potentials are dense in C(X) in the USC setting, while under specification for Z-actions non-freezing potentials form a dense G_δ, and provides insights into the analyticity of the pressure and rapid phase-switching phenomena. These results extend thermodynamic formalism to zero-temperature limits in broad dynamical contexts and furnish constructive tools for engineering freezing behavior across symbolic and smooth systems, with implications for understanding phase transitions and ground-state selection. In particular, the analysis shows that the zero-temperature asymptotics of the modified pressure P_φ(β) are governed by the slant asymptote f(β) = β Max(φ) + h_∞(φ), and that freezing corresponds to achieving this asymptote at some β_0, often accompanied by non-analyticity signaling a phase transition.
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 $φ$, there exists some inverse temperature $β_0 > 0$ such that for all $α, β> β_0$, the collection of equilibrium states for $αφ$ and $βφ$ coincide. In this sense, below the temperature $1 / β_0$, the system "freezes" on a fixed collection of equilibrium states. We show that for a given invariant measure $μ$, it is no more restrictive that $μ$ is the freezing state for some potential than it is for $μ$ 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 $μ$ 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 $G_δ$.