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Quantitative recurrence and large deviations for Teichmüller geodesic flow. (English) Zbl 1108.32007
Geom. Dedicata 119, 121-140 (2006); erratum ibid. 131, 231 (2008).
Let $$Q_g$$ be the moduli space of unit-area holomorphic quadratic differentials on a compact Riemann surface of genus $$g\geq 2$$. It is stratified by integer partitions of $$4g-4$$. Suppose that $$Q$$ is a connected component of the strata. Let $$K$$ and $$A$$ be the standard subgroups of SL$$(2,\mathbb R)$$ acting on $$Q$$. The elements of $$K$$ are rotations $$r_{\theta}$$ $$(0 \leq \theta < 2\pi)$$ and the elements of $$A$$ are diagonal matrices $$g_t$$ with entries $$e^t$$ and $$e^{-t}$$ $$(t\in \mathbb R)$$. The action of $$K$$ is the circle flow, and that of $$A$$ is the Teichmüller geodesic flow. For $$q\in Q$$, the set $$Kq=\{r_{\theta} q: 0 \leq \theta < 2\pi \}$$ has a natural probability measure $$\nu$$ coming from the Haar measure on $$K$$. The author considers recurrence behavior of geodesic trajectories $$\{ g_t r_{\theta} q \}_{t \geq 0}$$, and estimates the measure of the set of angles. Precisely, let $$V:Q \to \mathbb R^+$$ be a proper continuous function. Put $$C_l=\{q: V(q) \leq l \}$$. Then it is proved that for all $$l$$ sufficiently large and all $$q\not \in C_l$$, there exist positive constants $$c_1(l,q), c_2(l)$$ such that $$\nu \{ \theta: g_t r_{\theta} q \not \in C_l, 0 \leq t \leq T\} \leq c_1 e^{-c_2 T}$$ for all $$T$$ sufficiently large. It is also proved that the probability that a random geodesic trajectory does not enter $$C_l$$ in $$S \leq t \leq T$$ decays exponentially in $$T$$, and that the so-called large deviations result for Teichmüller flow holds. From these results the author gives a conjecture on the deviation of ergodic averages for the billiard flow for rational-angle Euclidean polygons.
Also, the author gives several results on a random walk $$\{ X_n \}$$, where $$X_0=q$$ and the following points are chosen from $$\{ g_{\tau} r_{\theta} q: 0 \leq \theta < 2\pi \}$$ for a fixed $$\tau >0$$ and $$q\in Q$$. One of them is that the probability of $$X_i \not \in C_l$$ $$(1\leq i \leq n)$$ decreases exponentially as $$n$$ grows.
The statements are studied more precisely for the proper functions $$V_{\delta}$$ $$(0< \delta <1)$$ and the compact sets $$C_{\delta,l}=\{ q: V_{\delta}(q)\leq l \}$$.

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations 30F60 Teichmüller theory for Riemann surfaces
##### Keywords:
moduli spaces; geodesic flow; large deviations
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##### References:
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