Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of Abelian differentials

Bufetov, Alexander

doi:10.1090/S0894-0347-06-00528-5

Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of Abelian differentials
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by Alexander I. Bufetov

J. Amer. Math. Soc. 19 (2006), 579-623

DOI: https://doi.org/10.1090/S0894-0347-06-00528-5

Published electronically: February 22, 2006

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Abstract:

The aim of this paper is to prove a stretched-exponential bound for the decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations. A corollary is the Central Limit Theorem for the Teichmüller flow on the moduli space of abelian differentials with prescribed singularities.

References

William A. Veech, Projective Swiss cheeses and uniquely ergodic interval exchange transformations, Ergodic theory and dynamical systems, I (College Park, Md., 1979–80), Progr. Math., vol. 10, Birkhäuser, Boston, Mass., 1981, pp. 113–193. MR 633764
William A. Veech, Interval exchange transformations, J. Analyse Math. 33 (1978), 222–272. MR 516048, DOI 10.1007/BF02790174
William A. Veech, Projective Swiss cheeses and uniquely ergodic interval exchange transformations, Ergodic theory and dynamical systems, I (College Park, Md., 1979–80), Progr. Math., vol. 10, Birkhäuser, Boston, Mass., 1981, pp. 113–193. MR 633764
Anton Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 325–370 (English, with English and French summaries). MR 1393518, DOI 10.5802/aif.1517
V. I. Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR 168 (1966), 1009–1011 (Russian). MR 0199347
Gérard Rauzy, Échanges d’intervalles et transformations induites, Acta Arith. 34 (1979), no. 4, 315–328 (French). MR 543205, DOI 10.4064/aa-34-4-315-328
M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739–741 (Russian). MR 0251785
Carlangelo Liverani, Central limit theorem for deterministic systems, International Conference on Dynamical Systems (Montevideo, 1995) Pitman Res. Notes Math. Ser., vol. 362, Longman, Harlow, 1996, pp. 56–75. MR 1460797
Michael Keane, Interval exchange transformations, Math. Z. 141 (1975), 25–31. MR 357739, DOI 10.1007/BF01236981
Giovanni Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), no. 1, 1–103. MR 1888794, DOI 10.2307/3062150
Lai-Sang Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153–188. MR 1750438, DOI 10.1007/BF02808180
Véronique Maume-Deschamps, Projective metrics and mixing properties on towers, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3371–3389. MR 1828610, DOI 10.1090/S0002-9947-01-02786-6
Ja. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421
L. A. Bunimovich and Ya. G. Sinaĭ, Statistical properties of Lorentz gas with periodic configuration of scatterers, Comm. Math. Phys. 78 (1980/81), no. 4, 479–497. MR 606459, DOI 10.1007/BF02046760
Ian Melbourne and Andrei Török, Statistical limit theorems for suspension flows, Israel J. Math. 144 (2004), 191–209. MR 2121540, DOI 10.1007/BF02916712
A. N. Kolmogorov, A local limit theorem for classical Markov chains, Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 281–300 (Russian). MR 0031216

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Caroline Series, The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), no. 1, 69–80. MR 810563, DOI 10.1112/jlms/s2-31.1.69
Caroline Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergodic Theory Dynam. Systems 6 (1986), no. 4, 601–625. MR 873435, DOI 10.1017/S0143385700003722
S. P. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Ergodic Theory Dynam. Systems 5 (1985), no. 2, 257–271. MR 796753, DOI 10.1017/S0143385700002881
Howard Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), no. 1, 169–200. MR 644018, DOI 10.2307/1971341
J. L. Doob, Stochastic processes, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. Reprint of the 1953 original; A Wiley-Interscience Publication. MR 1038526
V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499–530 (Russian). MR 0143873
V. A. Rohlin, New progress in the theory of transformations with invariant measure. , Russian Math. Surveys 15 (1960), no. 4, 1–22. MR 0132155, DOI 10.1070/RM1960v015n04ABEH004095
Ya. G. Sinaĭ, Topics in ergodic theory, Princeton Mathematical Series, vol. 44, Princeton University Press, Princeton, NJ, 1994. MR 1258087
Maxim Kontsevich and Anton Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), no. 3, 631–678. MR 2000471, DOI 10.1007/s00222-003-0303-x
I. P. Kornfel′d, Ya. G. Sinaĭ, and S. V. Fomin, Ergodicheskaya teoriya, “Nauka”, Moscow, 1980 (Russian). MR 610981
John Hubbard and Howard Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), no. 3-4, 221–274. MR 523212, DOI 10.1007/BF02395062
M. Kontsevich, Lyapunov exponents and Hodge theory, The mathematical beauty of physics (Saclay, 1996) Adv. Ser. Math. Phys., vol. 24, World Sci. Publ., River Edge, NJ, 1997, pp. 318–332. MR 1490861