Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-29T21:07:04.643Z Has data issue: false hasContentIssue false
  • English
  • Français

Decay of correlations for non-uniformly expanding systems with general return times

Published online by Cambridge University Press:  28 January 2013

IAN MELBOURNE  and
DALIA TERHESIU
IAN MELBOURNE
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK (email: i.melbourne@warwick.ac.uk)
DALIA TERHESIU
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK (email: i.melbourne@warwick.ac.uk)
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a unified treatment of decay of correlations for non-uniformly expanding systems with a good inducing scheme. In addition to being more elementary than previous treatments, our results hold for general integrable return time functions under fairly mild conditions on the inducing scheme.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 3 , June 2014 , pp. 893 - 918
Copyright
©2013 Cambridge University Press 

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]Aaronson, J., Denker, M., Sarig, O. and Zweimüller, R.. Aperiodicity of cocycles and conditional local limit theorems. Stoch. Dyn. 4 (2004), 31–62.Google Scholar
[2]Araújo, V., Luzzatto, S. and Viana, M.. Invariant measure for interval maps with critical points and singularities. Adv. Math. 221 (2009), 1428–1444.Google Scholar
[3]Chazottes, J.-R. and Gouëzel, S.. Optimal concentration inequalities for dynamical systems. Commun. Math. Phys. 316 (2012), 843–889.Google Scholar
[4]Feller, W.. An Introduction to Probability Theory and its Applications, II. Wiley, New York, 1966.Google Scholar
[5]Gouëzel, S.. Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139 (2004), 29–65.Google Scholar
[6]Gouëzel, S.. Vitesse de décorrélation et théorèmes limites pour les applications non uniformément dilatantes. PhD Thesis, Ecole Normale Supérieure, 2004.Google Scholar
[7]Gouëzel, S.. Private communication.Google Scholar
[8]Holland, M.. Slowly mixing systems and intermittency maps. Ergod. Th. & Dynam. Sys. 25 (2005), 133–159.Google Scholar
[9]Hu, H. and Vaienti, S.. Absolutely continuous invariant measures for non-uniformly expanding maps. Ergod. Th. & Dynam. Sys. 29 (2009), 1185–1215.Google Scholar
[10]Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19 (1999), 671–685.Google Scholar
[11]Luzzatto, S. and Melbourne, I.. Statistical properties and decay of correlations for interval maps with critical points and singularities. Commun. Math. Phys. to appear.Google Scholar
[12]Maume-Deschamps, V.. Projective metrics and mixing properties on towers. Trans. Amer. Math. Soc. 353 (2001), 3371–3389.Google Scholar
[13]Melbourne, I.. Decay of correlations for slowly mixing flows. Proc. Lond. Math. Soc. 98 (2009), 163–190.Google Scholar
[14]Melbourne, I. and Terhesiu, D.. Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. Math. 189 (2012), 61–110.Google Scholar
[15]Parry, W. and Pollicott, M.. Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics (Astérique, 187–188). Société Mathématique de France, Montrouge, 1990.Google Scholar
[16]Rychlik, M.. Bounded variation and invariant measures. Studia Math. 76 (1983), 69–80.CrossRefGoogle Scholar
[17]Sarig, O. M.. Subexponential decay of correlations. Invent. Math. 150 (2002), 629–653.Google Scholar
[18]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. (2) 147 (1998), 585–650.Google Scholar
[19]Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.Google Scholar
[20]Zweimüller, R.. Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points. Nonlinearity 11 (1998), 1263–1276.Google Scholar