Quantitative recurrence and hyperbolicity for the Teichmüller flow.

Ian Frankel (Toronto)

Monday 1st April 16:00-17:00 Maths 311B

Abstract

We will explain how quantitatively recurrent geodesics for the Teichmüller metric on the moduli space of Riemann surfaces behave like geodesics in a negatively curved space. More precisely,
 
Let \{(X_t,q_t): 0 \leq t \leq L\} be the family of quadratic differentials corresponding to a length L segment of a Teichmüller geodesic, parametrized by the metric. Let K be a compact subset of the moduli space of Riemann surfaces. If the set of times t for which X_t belongs to K has measure at least \theta L, and contains 0 and L, we will explain why the unit neighborhood of (X_0,q_0) in its strongly stable leaf is contracted by Ce^{-aL} in some metric, where the constants C,a depend on K and \theta but not L.

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