Rich phase transitions in step skew-products

TL;DR

This paper constructs skew product dynamical systems over a horseshoe that display complex phase transitions in topological pressure, linked to central Lyapunov exponents and coexistence of multiple equilibrium states.

Contribution

It introduces new examples of partially hyperbolic skew products with rich phase transition phenomena related to Lyapunov exponents and equilibrium states.

Findings

Existence of phase transitions in topological pressure.

Coexistence of multiple equilibrium states with positive entropy.

Presence of a measure of maximal entropy with nonpositive central exponent.

Abstract

We present examples of partially hyperbolic and topologically transitive local diffeomorphisms defined as skew products over a horseshoe which exhibit rich phase transitions for the topological pressure. This phase transition follows from a gap in the spectrum of the central Lyapunov exponents. It is associated to the coexistence of two equilibrium states with positive entropy. The diffeomorphisms mix hyperbolic behavior of different types. However, in some sense the expanding behavior is not dominating which is indicated by the existence of a measure of maximal entropy with nonpositive central exponent.