Phase transitions for surface diffeomorphisms

TL;DR

This paper investigates phase transitions in $C^1$ surface diffeomorphisms, showing that such transitions are generically present among non-Anosov diffeomorphisms and relate to non-hyperbolic periodic points.

Contribution

It establishes that phase transitions are $C^1$-Baire generic in non-Anosov surface diffeomorphisms and characterizes their occurrence in $C^2$ systems with dominated splittings.

Findings

Phase transitions are $C^1$-Baire generic among non-Anosov diffeomorphisms.

On non-torus surfaces, generic $C^1$ diffeomorphisms exhibit phase transitions.

In $C^2$ systems with dominated splitting, phase transitions occur iff there are non-hyperbolic periodic points.

Abstract

In this paper we consider $C^1$ surface diffeomorphisms and study the existence of phase transitions, here expressed by the non-analiticity of the pressure function associated to smooth and geometric-type potentials. We prove that the space of $C^1$-surface diffeomorphisms admitting phase transitions is a $C^1$-Baire generic subset of the space of non-Anosov diffeomorphisms. In particular, if $S$ is a compact surface which is not homeomorphic to the 2-torus then a $C^1$-generic diffeomorphism on $S$ has phase transitions. We obtain similar statements in the context of $C^1$--volume preserving diffeomorphisms. Finally, we prove that a $C^2$-surface diffeomorphism exhibiting a dominated splitting admits phase transitions if and only if has some non-hyperbolic periodic point.