Logarithmic laws and unique ergodicity

Jon Chaika; Rodrigo Treviño

doi:10.3934/jmd.2017022

Article Contents

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

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Logarithmic laws and unique ergodicity

Jon Chaika^1, and
Rodrigo Treviño^2,

1.
Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090, USA
2.
Department of Mathematics, Brooklyn College, City University of New York, Brooklyn, NY 11210-2889, USA

Received: June 07, 2017

Revised: August 24, 2017

Published: November 2017

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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.

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Mathematics Subject Classification: Primary: 30F60; Secondary: 37A25.

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Figure 1. The times $s(\delta)$ and $s(\delta')$ are both smaller than the return times of $I_a$ and $I_c$, respectively. Since $|I_b|\leq \frac{\mathrm{Area(subcomplex)}}{\text{return time to }I_b}$, and so, if the return time to $I_b$ is much greater than $\max\{s(\delta), s(\delta')\}$, we have a saddle connection between $p$ and $p'$ that is made small under $g_t$.

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Figure 2. On the left, an illustration of the definition of shadowing. On the right, the first step in the recursive procedure used in the Proof of Lemma 24.

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