Journal of Modern Dynamics (JMD)

Dynamics of the Teichmüller flow on compact invariant sets

Pages: 393 - 418, Issue 2, April 2010      doi:10.3934/jmd.2010.4.393

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Ursula Hamenstädt - Mathematisches Institut der Rheinischen Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany (email)

Abstract: Let $S$ be an oriented surface of genus $g\geq 0$ with $m\geq 0$ punctures and $3g-3+m\geq 2$. For a compact subset $K$ of the moduli space of area-one holomorphic quadratic differentials for $S$, let $\delta(K)$ be the asymptotic growth rate of the number of periodic orbits for the Teichmüller flow $\Phi^t$ which are contained in $K$. We relate $\delta(K)$ to the topological entropy of the restriction of $\Phi^t$ to $K$. Moreover, we show that sup$_K\delta(K)=6g-6+2m$.

Keywords:  Teichmüller flow, expansive, Anosov closing, orbit counting, entropy
Mathematics Subject Classification:  Primary: 37A35; Secondary: 30F60.

Received: May 2010;      Available Online: August 2010.