Low-temperature phase transitions in the quadratic family

TL;DR

This paper presents the first example of a quadratic map with a phase transition after the first zero of the pressure function, revealing non-analytic behavior in related dimension spectra and rate functions.

Contribution

It introduces a specific non-uniformly hyperbolic quadratic map exhibiting a phase transition, contrasting with the hyperbolic case.

Findings

Existence of a quadratic map with a phase transition after the first zero of pressure

Non-analyticity in dimension spectra and large deviation rate functions

The map has a non-recurrent critical point, indicating strong non-uniform hyperbolicity

Abstract

We give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function. This implies that several dimension spectra and large deviation rate functions associated to this map are not (expected to be) real analytic, in contrast to the uniformly hyperbolic case. The quadratic map we study has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.