Journal of Modern Dynamics (JMD)

Logarithmic laws and unique ergodicity

Pages: 563 - 588, Volume 11, 2017      doi:10.3934/jmd.2017022

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Jon Chaika - Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090 , United States (email)
Rodrigo Treviño - Department of Mathematics, Brooklyn College, City University of New York, Brooklyn, NY 11210-2889, United States (email)

Abstract: We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodic properties of translation flow than hyperbolic geometry.

Keywords:  Unique ergodicity, translation surfaces, logarithm laws.
Mathematics Subject Classification:  Primary: 30F60; Secondary: 37A25.

Received: June 2017;      Revised: August 2017;      Available Online: November 2017.