The K-Property for Some Unique Equilibrium States in Flows and Homeomorphisms

TL;DR

This paper establishes criteria for the K-property in flows and homeomorphisms, introducing new techniques and decompositions, and applies these to specific dynamical systems like Mañé diffeomorphisms and the Katok map.

Contribution

It refines previous assumptions for proving the K-property in flows and introduces discrete-time results and new decompositions, expanding the toolkit for dynamical systems analysis.

Findings

Criteria for the K-property in flows and homeomorphisms

Introduction of one-sided λ-decompositions

Application to Mañé diffeomorphisms and the Katok map

Abstract

We set out some general criteria to prove the K-property, refining the assumptions used in arXiv:1906.09315 for the flow case, and introducing the analogous discrete-time result. We also introduce one-sided $\lambda$-decompositions, as well as multiple techniques for checking the pressure gap required to show the K-property. We apply our results to the family of Ma\~n\'e diffeomorphisms and the Katok map. Our argument builds on the orbit decomposition theory of Climenhaga and Thompson.