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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Abstract

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.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624045

Digital Object Identifier
doi:10.1214/16-AIHP758

Mathematical Reviews number (MathSciNet)
MR3689971

Zentralblatt MATH identifier
1379.37012

Subjects
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

Keywords
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

Citation

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. https://projecteuclid.org/euclid.aihp/1500624045


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