Equilibrium states of weakly hyperbolic one-dimensional maps for Holder potentials

TL;DR

This paper proves that for certain weakly hyperbolic one-dimensional maps, all Holder continuous potentials are hyperbolic, leading to unique equilibrium states with exponential decay of correlations and no phase transitions.

Contribution

It establishes that weak hyperbolicity in one-dimensional maps ensures all Holder potentials are hyperbolic, a novel result in thermodynamic formalism.

Findings

All Holder continuous potentials are hyperbolic under weak hyperbolicity.

The pressure function is real analytic on the space of Holder functions.

Unique equilibrium states exhibit exponential decay of correlations.

Abstract

There is a wealth of results in the literature on the thermodynamic formalism for potentials that are, in some sense, "hyperbolic". We show that for a sufficiently regular one-dimensional map satisfying a weak hyperbolicity assumption, every Holder continuous potential is hyperbolic. A sample consequence is the absence of phase transitions: The pressure function is real analytic on the space of Holder continuous functions. Another consequence is that every Holder continuous potential has a unique equilibrium state, and that this measure has exponential decay of correlations.