Equilibrium states for a class of skew-products

TL;DR

This paper investigates equilibrium states for a class of partially hyperbolic skew-product systems on surfaces and tori, establishing conditions for their existence, uniqueness, and stability.

Contribution

It provides new sufficient conditions for the existence and uniqueness of equilibrium states in skew-products semi-conjugate to Axiom A systems.

Findings

Existence of equilibrium states under specified conditions

Uniqueness of equilibrium states for the class of skew-products

Discussion of statistical stability of these states

Abstract

We consider skew-products on $M\times \mathbb{T}^2$, where $M$ is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. This class of dynamics includes the open sets of $\Omega$-non-stable systems introduced by Abraham, Smale and Shub. We present sufficient conditions, both on the skew-products and the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability.