Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T11:43:09.486Z Has data issue: false hasContentIssue false
  • English
  • Français

A large deviations bound for the Teichmüller flow on the moduli space of abelian differentials

Published online by Cambridge University Press:  20 July 2010

VÍTOR ARAÚJO  and
ALEXANDER I. BUFETOV
VÍTOR ARAÚJO
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68.530, 21.945-970 Rio de Janeiro, RJ, Brazil (email: [email protected])
ALEXANDER I. BUFETOV
Affiliation:
Department of Mathematics, Rice University, MS 136, 6100 Main Street, Houston, Texas 77251-1892, USA The Steklov Institute of Mathematics, Russian Academy of Sciences, Gubkina str. 8, 119991 Moscow, Russia (email: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Large deviation rates are obtained for suspension flows over symbolic dynamical systems with a countable alphabet. We use a method employed previously by the first author [Large deviations bound for semiflows over a non-uniformly expanding base. Bull. Braz. Math. Soc. (N.S.)38(3) (2007), 335–376], which follows that of Young [Some large deviation results for dynamical systems. Trans. Amer. Math. Soc.318(2) (1990), 525–543]. As a corollary of the main results, we obtain a large deviation bound for the Teichmüller flow on the moduli space of abelian differentials, extending earlier work of Athreya [Quantitative recurrence and large deviations for Teichmuller geodesic flow. Geom. Dedicata119 (2006), 121–140].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011 , pp. 1043 - 1071
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Araújo, V.. Large deviations bound for semiflows over a non-uniformly expanding base. Bull. Braz. Math. Soc. (N.S.) 38(3) (2007), 335–376.CrossRefGoogle Scholar
[2]Athreya, J. S.. Quantitative recurrence and large deviations for Teichmüller geodesic flow. Geom. Dedicata 119 (2006), 121–140.CrossRefGoogle Scholar
[3]Avila, A., Gouëzel, S. and Yoccoz, J.-C.. Decay of correlations for Teichmüller flows. Publ. Math. Inst. Hautes Études Sci. 104 (2006), 143–211.CrossRefGoogle Scholar
[4]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
[5]Bufetov, A. and Gurevich, B.. Existence and uniqueness of the measure of maximal entropy for the Teichmüller flow on the moduli space of abelian differentials. Preprint, 2008, arXiv.org.CrossRefGoogle Scholar
[6]Bufetov, A. I.. 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. J. Amer. Math. Soc. 19(3) (2006), 579–623.CrossRefGoogle Scholar
[7]Buzzi, J. and Sarig, O.. Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Ergod. Th. & Dynam. Sys. 23(5) (2003), 1383–1400.CrossRefGoogle Scholar
[8]Cornfeld, I. P., Fomin, S. V. and Sinaĭ, Y. G.. Ergodic Theory (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245). Springer, New York, 1982; translated from the Russian by A. B. Sosinskiĭ.CrossRefGoogle Scholar
[9]Forni, G.. Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. of Math. (2) 155(1) (2002), 1–103.CrossRefGoogle Scholar
[10]Gurevich, B. M. and Savchenko, S. V.. Thermodynamic formalism for symbolic Markov chains with a countable number of states. Uspekhi Mat. Nauk 53(2(320)) (1998), 3–106.Google Scholar
[11]Hubbard, J. and Masur, H.. Quadratic differentials and foliations. Acta Math. 142(3–4) (1979), 221–274.CrossRefGoogle Scholar
[12]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
[13]Kontsevich, M.. Lyapunov Exponents and Hodge Theory. The Mathematical Beauty of Physics: A Memorial Volume for Claude Itzykson (Saclay, France, 5–7 June 1996) (Advanced Series in Mathematical Physics, 24). Eds. Drouffe, J. M. and Zuber, J. B.. World Scientific, Singapore, 1997, pp. 318–332.Google Scholar
[14]Kontsevich, M. and Zorich, A.. Connected components of the moduli spaces of abelian differentials with prescribed singularities. Invent. Math. 153(3) (2003), 631–678.CrossRefGoogle Scholar
[15]Masur, H.. Interval exchange transformations and measured foliations. Ann. of Math. (2) 115(1) (1982), 169–200.CrossRefGoogle Scholar
[16]Rokhlin, V. A.. On the fundamental ideas of measure theory. Amer. Math. Soc. Transl. 10 (1962), 1–52; translated from Mat. Sb. 25 (1949), 107–150.Google Scholar
[17]Rokhlin, V. A.. Lectures on the entropy theory of transformations with invariant measure. Russian Math. Surveys 22(5) (1967), 1–52; translated from Uspekhi Mat. Nauk 22(5) (1967), 3–56.CrossRefGoogle Scholar
[18]Sarig, O. M.. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19(6) (1999), 1565–1593.CrossRefGoogle Scholar
[19]Veech, W. A.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115(1) (1982), 201–242.CrossRefGoogle Scholar
[20]Veech, W. A.. The Teichmüller geodesic flow. Ann. of Math. (2) 124(3) (1986), 441–530.CrossRefGoogle Scholar
[21]Waddington, S.. Large deviations asymptotics for Anosov flows. Ann. Inst. H. Poincaré Sect. C 13(4) (1996), 445–484.CrossRefGoogle Scholar
[22]Walters, P.. An Introduction to Ergodic Theory. Springer, Berlin, 1982.CrossRefGoogle Scholar
[23]Young, L. S.. Some large deviation results for dynamical systems. Trans. Amer. Math. Soc. 318(2) (1990), 525–543.Google Scholar
[24]Zorich, A.. Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier (Grenoble) 46(2) (1996), 325–370.CrossRefGoogle Scholar