# American Institute of Mathematical Sciences

April  2010, 4(2): 393-418. doi: 10.3934/jmd.2010.4.393

## Dynamics of the Teichmüller flow on compact invariant sets

 1 Mathematisches Institut der Rheinischen Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

Received  May 2010 Published  August 2010

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$_K\delta(K)=6g-6+2m$.
Citation: Ursula Hamenstädt. Dynamics of the Teichmüller flow on compact invariant sets. Journal of Modern Dynamics, 2010, 4 (2) : 393-418. doi: 10.3934/jmd.2010.4.393
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