# American Institute of Mathematical Sciences

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

## Logarithmic laws and unique ergodicity

 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

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.

Citation: Jon Chaika, Rodrigo Treviño. Logarithmic laws and unique ergodicity. Journal of Modern Dynamics, 2017, 11: 563-588. doi: 10.3934/jmd.2017022
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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 $g_t$.
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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