# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Avila, A., Gouezel, S., Yoccoz, J.-C.: Exponential mixing for thev Teichmuller flow. Preprint, 2005. arxiv.org/math.DS/0511614  Bufetov, A.: Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmueller flow on the moduli space of Abelian differentials. preprint, 2005. arxiv.org/math.DS/0506222  Cheung Y. (2004). Slowly divergent geodesics in moduli space. Conform. Geom. Dynam. 8:167–189 · Zbl 1067.37036 · doi:10.1090/S1088-4173-04-00113-4  Cheung, Y., Masur, H.: A divergent Teichmuller geodesic with uniquely ergodic vertical foliation. Preprint, 2005, arxiv.org/math.DS/0501296  Cheung, Y., Masur, H.: Minimal nonergodic directions on genus 2 translation surfaces. To appear in Ergodic Theory Dynam. Systems (2005) arxiv.org/math.DS/0501286 · Zbl 1087.37008  Durrett, R.: Probability: Theory and Examples, 2nd edn. Duxbury Press (1996) · Zbl 1202.60001  Eskin, A., Margulis, G.: Recurrence properties of random walks on finite volume homogeneous manifolds. In: Random Walks and Geometry, Walter de Gruyter GmbH and Co. KG, Berlin pp. 431–444 (2004). · Zbl 1064.60092  Eskin A., Masur H. (2001). Asymptotic Formulas on Flat Surfaces. Ergodic Theory Dynam. Systems 21:443–478 · Zbl 1096.37501 · doi:10.1017/S0143385701001225  Evans, L.C., Gariepy, R.: Measure Theory and Fine Properties of Functions. CRC Press (1991) · Zbl 0804.28001  Farb, B., Margalit, D.: A primer on mapping class groups, in preparation · Zbl 1245.57002  Forni G. (2002). Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. Math. 155(1):1–103 · Zbl 1034.37003 · doi:10.2307/3062150  Kerckhoff S., Masur H., Smillie J. (1986). Ergodicity of billiard flows and quadratic differentials. Ann. Math. 124(2):293–311 · Zbl 0637.58010 · doi:10.2307/1971280  Kleinbock D.Y., Margulis G.A. (1999). Logarithm laws for flows on homogeneous spaces. Invent. Math. 138:451–494 · Zbl 0934.22016 · doi:10.1007/s002220050350  Kleinbock D., Weiss B. (2004). Bounded geodesics in moduli space. Int. Math. Res. Not. 30:1551–1560 · Zbl 1075.37008 · doi:10.1155/S1073792804133412  Kontsevich M., Zorich A. (2003). Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math. 153(3):631–678 · Zbl 1087.32010 · doi:10.1007/s00222-003-0303-x  Masur H. (1982). Interval exchange transformations and measured foliations. Ann. Math. 115:169–200 · Zbl 0497.28012 · doi:10.2307/1971341  Masur H. (1993). Logarithm law for geodesics in moduli space. Contemp Math. 150:229–245 · Zbl 0790.32022  Masur, H., Tabachnikov, S.: Rational billiards and flat structures. Handbook of Dynamical Systems, Vol. 1A, pp. 1015–1089. North-Holland, Amsterdam (2002) · Zbl 1057.37034  Minsky Y., Weiss B. (2002). Non-divergence of horocyclic flows on moduli space. J. Reine Angew. Math. 552:131–177 · Zbl 1079.32011 · doi:10.1515/crll.2002.088  Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer-Verlag (1993) · Zbl 0925.60001  Sullivan D. (1982). Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math. 149:215–238 · Zbl 0517.58028 · doi:10.1007/BF02392354  Strebel, K.: Quadratic Differentials. Springer-Verlag (1984) · Zbl 0547.30001  Varadhan, S.R.S.: Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics Vol. 46. SIAM (1984) · Zbl 0549.60023  Veech W. (1982). Gauss measures for transformations on the space of interval exchange maps. Ann. Math. 115:201–242 · Zbl 0486.28014 · doi:10.2307/1971391  Veech W. (1986). Teichmuller geodesic flow. Ann. Math. 124:441–530 · Zbl 0658.32016 · doi:10.2307/2007091  Zemljakov A., Katok A. (1975). Topological transitivity of billiards in polygons. Mat. Zametki 19(2):291–300
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.