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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 \(C_{\delta,l}=\{ q: V_{\delta}(q)\leq l \}\).

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
Full Text: DOI
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