Renormalisation-induced phase transitions for unimodal maps

TL;DR

This paper investigates phase transitions in the thermodynamical formalism for unimodal maps, revealing multiple phase transitions for quadratic maps and a universal phase transition in the negative spectrum for all positive entropy unimodal quadratic maps.

Contribution

It demonstrates the occurrence of multiple phase transitions in quadratic maps and establishes a universal phase transition in the negative spectrum for all positive entropy unimodal quadratic maps.

Findings

Multiple phase transitions can occur at positive t for some quadratic maps.

All unimodal quadratic maps with positive topological entropy have a phase transition in the negative spectrum.

The study connects renormalisation with phase transition phenomena in dynamical systems.

Abstract

The thermodynamical formalism is studied for renormalisable maps of the interval and the natural potential $-t \log|Df|$. Multiple and indeed infinitely many phase transitions at positive $t$ can occur for some quadratic maps. All unimodal quadratic maps with positive topological entropy exhibit a phase transition in the negative spectrum.