Cohomological equation for geodesic flows on flat surfaces

TL;DR

This paper investigates solutions to the cohomological equation for geodesic flows on flat surfaces, establishing ergodicity and cohomology-free properties under specific geometric and Diophantine conditions.

Contribution

It proves the existence of solutions to the cohomological equation and demonstrates ergodicity and cohomology-free properties for certain flat surfaces with cone points.

Findings

Existence of solutions to the cohomological equation on flat surfaces.

Ergodicity of the holonomy foliation for non-rational holonomy surfaces.

Cohomology-free property of the horizontal foliated Laplacian under Diophantine conditions.

Abstract

We prove the existence of solutions of the cohomological equation for the geodesic flow on the unit tangent bundle of a compact flat surface with finitely many cone points. We also prove the ergodicity of the holonomy foliation for surfaces with non-rational holonomy, and the cohomology-free property of the horizontal foliated Laplacian under a simultaneous Diophantine condition.