Equilibrium states at freezing phase transition in unimodal maps with flat critical point

TL;DR

This paper investigates the statistical properties and phase transition behavior of equilibrium states in unimodal maps with flat critical points, revealing conditions for exponential decay of correlations and convergence to invariant measures.

Contribution

It provides a detailed analysis of equilibrium states at freezing phase transitions in unimodal maps with flat critical points, including decay rates and convergence properties.

Findings

Exponential decay of correlations for inverse temperatures in (t^-, t^+)

Unique equilibrium state at the transition when the critical point is not too flat

Weak convergence of measures to the acip at the freezing point

Abstract

An $S$-unimodal map $f$ with flat critical point satisfying the Misiurewicz condition displays a freezing phase transition in positive spectrum. We analyze statistical properties of the equilibrium state $\mu_t$ for the potential $-t\log|Df|$, as well as how the phase transition slows down the rate of decay of correlations. We show that $\mu_t$ has exponential decay of correlations for all inverse temperature $t$ contained in the positive entropy phase $(t^-,t^+)$. If the critical point is not too flat, then the freezing point $t^+$ is equal to $1$, and the absolutely continuous invariant probability measure (acip for short) is the unique equilibrium state at the transition. We exhibit a case in which the acip has sub-exponential decay of correlations and $\mu_t$ converges weakly to the acip as $t\nearrow t^+$.