Anosov diffeomorphisms of open surfaces

TL;DR

This paper investigates the existence of Anosov diffeomorphisms on open surfaces, establishing conditions under which such systems possess invariant measures along stable and unstable foliations.

Contribution

It demonstrates that under certain assumptions, Anosov diffeomorphisms on open surfaces admit a system of Margulis measures that are holonomy and dynamically invariant.

Findings

Existence of Margulis measures for Anosov diffeomorphisms on open surfaces

Conditions for density of periodic points and uniform geometry

Invariant measures along stable and unstable leaves

Abstract

We study the existence of Anosov diffeomorphisms on complete open surfaces. We show that under the assumptions of density of periodic points and uniform geometry that such diffeomorphisms have a system of Margulis measures, which are a holonomy invariant and dynamically invariant system of measures along the stable and unstable leaves.