Sensitive dependence of geometric Gibbs states at positive temperature

TL;DR

This paper presents the first example of smooth quadratic-like maps with sensitive dependence of geometric Gibbs states at positive temperature, demonstrating non-convergence and robustness in a new class of dynamical systems.

Contribution

It introduces the first known examples of non-convergence of geometric Gibbs states at positive temperature in smooth quadratic-like maps, answering a longstanding question.

Findings

Geometric Gibbs states do not converge at positive temperature for a dense set of maps.

Sensitive dependence of Gibbs states is robust in an open set of quadratic-like maps.

First examples of non-convergence at positive temperature in statistical mechanics.

Abstract

We give the first example of a smooth family of real and complex maps having sensitive dependence of geometric Gibbs states at positive temperature. This family consists of quadratic-like maps that are non-uniformly hyperbolic in a strong sense. We show that for a dense set of maps in the family the geometric Gibbs states do not converge at positive temperature. These are the first examples of non-convergence at positive temperature in statistical mechanics or the thermodynamic formalism, and answers a question of van Enter and Ruszel. We also show that this phenomenon is robust: There is an open set of analytic 2-parameter families of quadratic-like maps that exhibit sensitive dependence of geometric Gibbs states at positive temperature.