Flat Surfaces with Finite Holonomy Groups

TL;DR

This paper proves ergodicity of geodesic flows on flat surfaces with finite holonomy and extends this result to certain billiard flows, advancing understanding of dynamical systems on flat geometries.

Contribution

It establishes ergodicity for geodesic flows on flat surfaces with finite holonomy and applies this to billiard flows, a novel extension in flat surface dynamics.

Findings

Geodesic flows are ergodic on generic flat surfaces with finite holonomy.

Billiard flows on certain flat surfaces with boundary are also ergodic.

The results connect flat surface geometry with dynamical ergodic properties.

Abstract

We prove that flow of a generic geodesic on a flat surface with finite holonomy group is ergodic. We use this result to prove that flows of generic billiards on certain flat surfaces with boundary are also ergodic.