Every flat surface is Birkhoff and Oseledets generic in almost every direction

TL;DR

This paper demonstrates that for almost every direction on any flat surface, the Birkhoff and Oseledets theorems hold, using ergodic theory, rigidity results, and recent advances in cocycle continuity.

Contribution

It establishes the universality of Birkhoff and Oseledets genericity for all flat surfaces in almost every direction, extending previous results with new ergodic and rigidity techniques.

Findings

Birkhoff ergodic theorem applies in almost every direction for flat surfaces.

Oseledets multiplicative ergodic theorem holds in almost every direction.

Utilizes recent rigidity results and cocycle continuity theorems.

Abstract

We prove that the Birkhoff pointwise ergodic theorem and the Oseledets multiplicative ergodic theorem hold for every flat surface in almost every direction. The proofs rely on the strong law of large numbers, and on recent rigidity results for the action of the upper triangular subgroup of SL(2,R) on the moduli space of flat surfaces. Most of the results also use a theorem about continuity of splittings of the Kontsevich-Zorich cocycle recently proved by S. Filip.