Maximal Entropy Measures for Piecewise Affine Surface Homeomorphisms

TL;DR

This paper investigates the entropy of piecewise affine surface homeomorphisms, establishing the existence of maximal entropy measures and bounds on periodic points, advancing understanding of surface dynamics.

Contribution

It introduces a novel approach using jump transformations and good returns to analyze entropy without relying on Markov partitions.

Findings

Existence of finitely many ergodic measures maximizing entropy

A multiplicative lower bound on the number of periodic points

Development of a new method avoiding Markov partitions

Abstract

We study the dynamics of piecewise affine surface homeomorphisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended as a step towards the understanding of surface diffeomorphisms. We proceed by building a jump transformation, using not first returns but carefully selected "good" returns to dispense with Markov partitions. We control these good returns through some entropy and ergodic arguments.