Equilibrium states for potentials with $\sup\phi - \inf\phi < \htop(f)$

TL;DR

This paper investigates equilibrium states for smooth interval maps with potentials whose range is bounded by the topological entropy, establishing existence, uniqueness, and analyticity of pressure under these conditions.

Contribution

It introduces an inducing scheme approach to prove equilibrium states for potentials with bounded range, extending prior results and emphasizing the importance of the bounded range condition.

Findings

Existence and uniqueness of equilibrium states under bounded range condition

Analyticity of the pressure function in this setting

Comparison with Perron-Frobenius operator methods

Abstract

In the context of smooth interval maps, we study an inducing scheme approach to prove existence and uniqueness of equilibrium states for potentials $\phi$ with he `bounded range' condition $\sup \phi - \inf \phi < \htop$, first used by Hofbauer and Keller. We compare our results to Hofbauer and Keller's use of Perron-Frobenius operators. We demonstrate that this `bounded range' condition on the potential is important even if the potential is H\"older continuous. We also prove analyticity of the pressure in this context.