Natural equilibrium states for multimodal maps

TL;DR

This paper investigates the thermodynamic formalism for a broad class of real multimodal maps, establishing existence, uniqueness, and properties of equilibrium states and phase transitions within this setting.

Contribution

It extends thermodynamic formalism to larger classes of multimodal maps, proving existence and uniqueness of equilibrium states and analyzing phase transitions.

Findings

Existence and uniqueness of equilibrium states for geometric potentials.

Complete characterization of phase transitions in the pressure function.

Results on the existence of absolutely continuous invariant measures.

Abstract

This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials $-t \log|Df|$, for the largest possible interval of parameters $t$. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained.