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

Jon Chaika; Alex Eskin

doi:10.3934/jmd.2015.9.1

Article Contents

2015, Volume 9: 1-23. Doi: 10.3934/jmd.2015.9.1

This issue Previous Article Next Article On the rigidity of Weyl chamber flows and Schur multipliers as topological groups

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

Jon Chaika^1, and
Alex Eskin^2,

1.
Department of Mathematics, University of Utah, 155 S. 1400 E., Room 233, Salt Lake City, UT 84112
2.
Department of Mathematics, University of Chicago, Chicago, IL 60637

Received: October 31, 2013

Revised: September 30, 2014

Published: April 2015

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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 $SL(2,\mathbb R)$ 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.

Keywords:

Ergodic theorems,
flat surfaces.

Mathematics Subject Classification: Primary: 37A10; Secondary: 37A25.

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