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Logarithmic laws and unique ergodicity

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  • 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.

    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 $g_t$.

    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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