Thermodynamic formalism for expanding measures

TL;DR

This paper develops a thermodynamic formalism for expanding measures in dynamical systems, proving uniqueness of equilibrium states for certain maps and potentials, with applications to quadratic and Viana maps.

Contribution

It establishes the uniqueness of equilibrium states among expanding measures for non-flat $C^{1+}$ maps and small oscillation potentials, extending thermodynamic formalism in this context.

Findings

Uniqueness of equilibrium states for expanding measures with H"older potentials.

No phase transition for Collet-Eckmann quadratic maps under H"older potentials.

Existence and uniqueness of equilibrium states for Viana maps with small oscillation potentials.

Abstract

In this paper we study the thermodynamic formalism of strongly transitive endomorphisms $f$, focusing on the set all expanding measures. In case $f$ is a non-flat $C^{1+}$ map defined on a Riemannian manifold, these are invariant probability measures with all its Lyapunov exponents positive. Given a H\"older continuous potential $\varphi$ we prove the uniqueness of the equilibrium state among the space of expanding measures. Moreover, we show that the existence of an expanding measure $\mu$ maximizing the entropy on the the space of expanding measures implies the existence and uniqueness of equilibrium state $\mu_{\varphi}$ on the space of expanding measures for any H\"older continuous potential $\varphi$ with a small oscillation $\text{osc }\varphi=\sup\varphi-\inf\varphi$. As some applications, we prove that Collet-Eckmann quadratic maps does not admit phase transition for H\"older potential, and show that for Viana maps and every H\"older continuous potential of sufficiently small oscillation has a unique equilibrium state.