# Quantitative recurrence and hyperbolicity for the Teichmüller flow.

### Ian Frankel (Toronto)

Monday 1st April, 2019 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 $L$.