Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Recurrence statistics for the space of interval exchange maps and the Teichmüller flow on the space of translation surfaces

Romain Aimino, Matthew Nicol, and Mike Todd

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In this paper we show that the transfer operator of a Rauzy–Veech–Zorich renormalization map acting on a space of quasi-Hölder functions is quasicompact and derive certain statistical recurrence properties for this map and its associated Teichmüller flow. We establish Borel–Cantelli lemmas, Extreme Value statistics and return time statistics for the map and flow. Previous results have established quasicompactness in Hölder or analytic function spaces, for example the work of M. Pollicott and T. Morita. The quasi-Hölder function space is particularly useful for investigating return time statistics. In particular we establish the shrinking target property for nested balls in the setting of Teichmüller flow. Our point of view, approach and terminology derive from the work of M. Pollicott augmented by that of M. Viana.


Dans cet article, nous démontrons que l’opérateur de transfert de l’application de renormalisation de Rauzy–Veech–Zorich est quasi-compact sur l’espace des fonctions quasi-Hölder, et nous en déduisons plusieurs propriétés de récurrence statistiques pour cette application et le flot de Teichmüller associé. Nous établissons des lemmes de Borel–Cantelli, des statistiques des valeurs extrêmes et des temps de retour pour l’application et le flot. De précédents résultats ont établi la quasi-compacité dans des espaces de fonctions Hölder ou analytiques, comme par exemple les travaux de M. Pollicott ou de T. Morita. L’espace fonctionnel quasi-Hölder est particulièrement adapté pour analyser les propriétés de récurrence statistiques. En particulier, nous démontrons la propriétés des cibles rétrécissantes pour des boules imbriquées dans le cadre du flot de Teichmüller. Notre point de vue, approche et terminologie proviennent du travail de M. Pollicott ainsi que de celui de M. Viana.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1371-1401.

Received: 25 September 2015
Revised: 15 March 2016
Accepted: 14 April 2016
First available in Project Euclid: 21 July 2017

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Mathematical Reviews number (MathSciNet)

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Primary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 60G70: Extreme value theory; extremal processes

Interval exchange map Teichmüller flow Rauzy–Veech–Zorich renormalisation map Transfer operator Borel–Cantelli lemmas Extreme Value Laws Return/hitting time statistics Quasi-Hölder function space


Aimino, Romain; Nicol, Matthew; Todd, Mike. Recurrence statistics for the space of interval exchange maps and the Teichmüller flow on the space of translation surfaces. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1371--1401. doi:10.1214/16-AIHP758.

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  • [1] J. Aaronson. An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs 50. AMS, Providence, 1997.
  • [2] M. Abadi and A. Galves. Inequalities for the occurrence times of rare events in mixing processes. The state of the art. Markov Process. Related Fields 7 (2001) 97–112.
  • [3] J. Athreya. Quantitative recurrence and large deviations for Teichmüller geodesic flow. Geom. Dedicata 119 (2006) 121–140.
  • [4] A. Avila and A. Bufetov. Exponential decay of correlations for the Rauzy–Veech–Zorich induction map. In Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow 203–211. Fields Inst. Commun. 51. Amer. Math. Soc., Providence, RI, 2007.
  • [5] A. Avila, S. Gouëzel and J.-C. Yoccoz. Exponential mixing of the Teichmüller flow. Publ. Math. Inst. Hautes Etudes Sci. 104 (2006) 143–211.
  • [6] H. Aytaç, J. M. Freitas and S. Vaienti. Laws of rare events for deterministic and random dynamical systems. Trans. Amer. Math. Soc. 367 (2015) 8229–8278.
  • [7] V. Baladi. Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics 16. World Scientific, River Edge, 2000.
  • [8] H. Bruin, B. Saussol, S. Troubetzkoy and S. Vaienti. Return time statistics via inducing. Ergodic Theory Dynam. Systems 23 (2003) 991–1013.
  • [9] A. Bufetov. 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 (2006) 579–623.
  • [10] A. I. Bufetov and B. M. Gurevich. Existence and uniqueness of a measure with maximal entropy for the Teichmüller flow on the moduli space of abelian differentials. Mat. Sb. 202 (2011) 3–42.
  • [11] N. Chernov and D. Kleinbock. Dynamical Borel–Cantelli lemmas for Gibbs measures. Israel J. Math. 122 (2001) 1–27.
  • [12] P. Collet. Statistics of closest return for some non-uniformly hyperbolic systems. Ergodic Theory Dynam. Systems 21 (2001) 401–420.
  • [13] D. Dolgopyat. Limit theorems for partially hyperbolic systems. Trans. Amer. Math. Soc. 356 (2004) 1637–1689.
  • [14] M. Einsiedler and T. Ward. Ergodic Theory with a View Towards Number Theory. Graduate Texts in Mathematics 259. Springer, London, 2011.
  • [15] A. C. M. Freitas, J. M. Freitas and M. Todd. Hitting time statistics and extreme value theory. Probab. Theory Related Fields 147 (2010) 675–710.
  • [16] A. C. M. Freitas, J. M. Freitas and M. Todd. The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics. Comm. Math. Phys. 321 (2013) 483–527.
  • [17] A. C. M. Freitas, J. M. Freitas, M. Todd and S. Vaienti. Rare events for the Manneville–Pomeau map. Stochastic Process. Appl. 126 (11) (2016) 3463–3479.
  • [18] S. Galatolo. Dimension and hitting time in rapidly mixing systems. Math. Res. Lett. 14 (5) (2007) 797–805.
  • [19] S. Galatolo and D. Kim. The dynamical Borel–Cantelli lemma and the waiting time problems. Indag. Math. (N.S.) 18 (3) (2007) 421–434.
  • [20] S. Gouëzel. Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139 (2004) 29–65.
  • [21] S. Gouëzel. A Borel–Cantelli lemma for intermittent interval maps. Nonlinearity 20 (6) (2007) 1491–1497.
  • [22] S. Gouëzel. Local limit theorem for nonuniformly partially hyperbolic skew-products and Farey sequences. Duke Math. J. 147 (2009) 192–284.
  • [23] C. Gupta. Extreme value distributions for some classes of non-uniformly partially hyperbolic dynamical systems. Ergodic Theory Dynam. Systems 30 (2010) 757–771.
  • [24] C. Gupta, M. Holland and M. Nicol. Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps. Ergodic Theory Dynam. Systems 31 (2011) 1363–1390.
  • [25] C. Gupta, M. Nicol and W. Ott. A Borel–Cantelli lemma for non-uniformly expanding dynamical systems. Nonlinearity 23 (2010) 1991–2008.
  • [26] N. Haydn, Y. Lacroix and S. Vaienti. Hitting and return time statistics in ergodic dynamical systems. Ann. Probab. 33 (2005) 2043–2050.
  • [27] N. Haydn, M. Nicol, T. Persson and S. Vaienti. A note on Borel–Cantelli lemmas for non-uniformly hyperbolic dynamical systems. Ergodic Theory Dynam. Systems 33 (2013) 475–498.
  • [28] N. T. Haydn, N. Winterberg and R. Zweimüller. Return-time statistics, hitting-time statistics and inducing. In Ergodic theory, Open Dynamics, and Coherent Structures 217–227. Springer Proceedings in Mathematics & Statistics 70. Springer, New York, 2014.
  • [29] H. Hennion. Sur un théorème spectral et son application aux noyaux lipschitziens. Proc. Amer. Math. Soc. 118 (1993) 627–634.
  • [30] M. Hirata. Poisson limit law for Axiom-A diffeomorphisms. Ergodic Theory Dynam. Systems 13 (1993) 533–556.
  • [31] M. Hirata, B. Saussol and S. Vaienti. Statistics of return times: A general framework and new applications. Comm. Math. Phys. 206 (1999) 33–55.
  • [32] M. P. Holland, M. Nicol and A. Török. Extreme value distributions for non-uniformly hyperbolic dynamical systems. Trans. Amer. Math. Soc. 364 (2012) 661–688.
  • [33] S. Kalikow and R. McCutcheon. An Outline of Ergodic Theory. Cambridge Studies in Advanced Mathematics 122. Cambridge University Press, Cambridge, 2010.
  • [34] M. Keane. Interval exchange transformations. Math. Z. 141 (1975) 25–31.
  • [35] G. Keller. Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrsch. Verw. Gebiete 69 (1985) 461–478.
  • [36] D. Kim. The dynamical Borel–Cantelli lemma for interval maps. Discrete Contin. Dyn. Syst. 17 (4) (2007) 891–900.
  • [37] D. Kleinbock and G. Margulis. Logarithm laws for flows on homogeneous spaces. Invent. Math. 138 (1999) 451–494.
  • [38] G. Lindgren, M. R. Leadbetter and H. Rootzén. Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. Springer-Verlag, New York-Berlin, 1983.
  • [39] L. Marchese. The Khinchin theorem for interval-exchange transformations. J. Mod. Dyn. 5 (2011) 123–183.
  • [40] L. Marchese. Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow. Bull. Soc. Math. France 140 (2013) 485–532.
  • [41] H. Masur. Interval exchange transformations and measured foliations. Ann. of Math. (2) 115 (1982) 169–200.
  • [42] H. Masur. Logarithm law for geodesics in moduli space. In Mapping Class Groups and Moduli Spaces of Riemann Surfaces 229–245. Comtemp. Math. 150. Amer. Math. Soc., Providence, RI, 1993.
  • [43] F. Maucourant. Dynamical Borel–Cantelli lemma for hyperbolic spaces. Israel J. Math. 152 (2006) 143–155.
  • [44] I. Melbourne and M. Nicol. Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260 (2005) 131–146.
  • [45] I. Melbourne and M. Nicol. Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360 (2008) 6661–6676.
  • [46] T. Morita. A generalized local limit theorem for Lasota–Yorke transformations. Osaka J. Math. 26 (1989) 579–595.
  • [47] T. Morita. Renormalized Rauzy inductions. Adv. Stud. Pure Math. 49 (2007) 263–288.
  • [48] W. Philipp. Some metrical theorems in number theory. Pacific J. Math. 20 (1967) 109–127.
  • [49] M. Pollicott. Statistical properties of the Rauzy–Veech–Zorich map. Unpublished notes. Available at
  • [50] O. Sarig. Subexponential decay of correlations. Invent. Math. 150 (2002) 629–653.
  • [51] B. Saussol. Absolutely continuous invariant measures for multidimensional expanding maps. Israel J. Math. 116 (2000) 223–248.
  • [52] W. Schmidt. A metrical theory in Diophantine approximation. Canad. J. Math. 12 (1960) 619–631.
  • [53] W. Schmidt. Metrical theorems on fractional parts of sequences. Trans. Amer. Math. Soc. 110 (1964) 493–518.
  • [54] V. G. Sprindzuk. Metric Theory of Diophantine Approximations. Scripta Series in Mathematics. V. H. Winston and Sons, Washington, D.C., 1979. Translated from the Russian and edited by Richard A. Silverman. With a foreword by Donald J. Newman.
  • [55] W. A. Veech. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982) 201–242.
  • [56] W. A. Veech. The Teichmüller geodesic flow. Ann. of Math. (2) 124 (1986) 441–530.
  • [57] M. Viana. Dynamics of interval exchange maps and Teichmüller flows. IMPA, 2008. Available at
  • [58] P. Walters. An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79. Springer, New York-Berlin, 1982.
  • [59] J.-C. Yoccoz. Continued fraction algorithms for interval exchange maps: An introduction. In Frontiers in Number Theory, Physics, and Geometry. I 401–435. Springer, Berlin, 2006.
  • [60] L.-S. Young. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147 (1998) 585–650.
  • [61] L.-S. Young. Recurrence times and rates of mixing. Israel J. Math. 110 (1999) 153–188.
  • [62] L. Zhang. Borel–Cantelli lemmas and extreme value theory for geometric Lorenz models. Nonlinearity 29 (1) (2016) 232–255.
  • [63] A. Zorich. Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier (Grenoble) 46 (1996) 325–370.