Abundant rich phase transitions in step skew products

TL;DR

This paper demonstrates the existence of complex phase transition phenomena in certain dynamical systems with rich intermingled hyperbolic behaviors, showing multiple non-differentiable points in the pressure function due to coexistence of distinct equilibrium states.

Contribution

It constructs examples of partially hyperbolic systems with step skew product dynamics exhibiting arbitrarily many rich phase transitions in their pressure functions.

Findings

Existence of systems with arbitrary many phase transitions.

Phase transitions linked to gaps in the central Lyapunov spectrum.

Coexistence of multiple equilibrium states with positive entropy.

Abstract

We study phase transitions for the topological pressure of geometric potentials of transitive sets. The sets considered are partially hyperbolic having a step skew product dynamics over a horseshoe with one-dimensional fibers corresponding to the central direction. The sets are genuinely non-hyperbolic containing intermingled horseshoes of different hyperbolic behavior (contracting and expanding center). We prove that for every $k\ge 1$ there is a diffeomorphism $F$ with a transitive set $\Lambda$ as above such that the pressure map $P(t)=P(t\, \varphi)$ of the potential $\varphi= -\log \,\lVert dF|_{E^c}\rVert$ ($E^c$ the central direction) defined on $\Lambda$ has $k$ rich phase transitions. This means that there are parameters $t_\ell$, $\ell=1,...,k$, where $P(t)$ is not differentiable and this lack of differentiability is due to the coexistence of two equilibrium states of $t_\ell\,\varphi$ with positive entropy and different Birkhoff averages. Each phase transition is associated to a gap in the central Lyapunov spectrum of $F$ on $\Lambda$.