Ergodic infinite group extensions of geodesic flows on translation surfaces

TL;DR

This paper demonstrates that most infinite group extensions of geodesic flows on translation surfaces are ergodic in nearly all directions, contrasting with some specific non-ergodic cases, and introduces a combinatorial method for analyzing such surfaces.

Contribution

It establishes the almost sure ergodicity of generic infinite group extensions of geodesic flows on translation surfaces, providing new insights and methods for their study.

Findings

Most infinite group extensions are ergodic in almost every direction.

Contrasts with non-ergodic staircases shown in prior work.

Introduces a combinatorial approach for analyzing square-tiled surfaces.

Abstract

We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. Recently K. Fr\c{a}czek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases. An appendix provides a combinatorial approach for the study of square-tiled surfaces.