On the Ergodicity of Flat Surfaces of Finite Area

TL;DR

This paper establishes ergodic theorems for finite-area flat surfaces, linking recurrence of Teichmuller orbits to ergodic properties of translation flows, with implications for surfaces of infinite genus and finite area.

Contribution

It introduces new ergodic theorems for flat surfaces with recurrent Teichmuller orbits and provides a criterion for ergodicity based on metric deformation control.

Findings

Ergodic theorems for flat surfaces with recurrent Teichmuller orbits.

A criterion for ergodicity based on deforming metrics.

Generalization of Cheung and Eskin's theorem for translation flows.

Abstract

We prove some ergodic theorems for flat surfaces of finite area. The first result concerns such surfaces whose Teichmuller orbits are recurrent to a compact subset of $SL(2;R)/SL(S)$, where $SL(S)$ is the Veech group of the surface. In this setting, this means that the translation flow on a flat surface can be renormalized through its Veech group to reveal ergodic properties of the translation flow. This result applies in particular to flat surfaces of infinite genus and finite area. Our second result is an criterion for ergodicity based on the control of deforming metric of a flat surface. Applied to translation flows on compact surfaces, it improves and generalizes a theorem of Cheung and Eskin.