Journal of Modern Dynamics (JMD)

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

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

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Jon Chaika - Department of Mathematics, University of Utah, 155 S. 1400 E., Room 233, Salt Lake City, UT 84112, United States (email)
Alex Eskin - Department of Mathematics, University of Chicago, Chicago, IL 60637, United States (email)

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.

Received: November 2013;      Revised: October 2014;      Available Online: May 2015.