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Upper and lower bounds for the correlation function via inducing with general return times

Published online by Cambridge University Press:  26 July 2016

HENK BRUIN  and
DALIA TERHESIU
HENK BRUIN
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar Morgensternplatz 1, 1090 Vienna, Austria email henk.bruin@univie.ac.at, daliaterhesiu@gmail.com
DALIA TERHESIU
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar Morgensternplatz 1, 1090 Vienna, Austria email henk.bruin@univie.ac.at, daliaterhesiu@gmail.com
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Abstract

For non-uniformly expanding maps inducing with a general return time to Gibbs Markov maps, we provide sufficient conditions for obtaining higher-order asymptotics for the correlation function in the infinite measure setting. Along the way, we show that these conditions are sufficient to recover previous results on sharp mixing rates in the finite measure setting for non-Markov maps, but for a larger class of observables. The results are illustrated by (finite and infinite measure-preserving) non-Markov interval maps with an indifferent fixed point.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 34 - 62
Copyright
© Cambridge University Press, 2016 

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References

Bingham, N. H., Goldie, C. M. and Teugels, J. L.. Regular Variation (Encyclopedia of Mathematics and its Applications, 27) . Cambridge University Press, Cambridge, 1987.Google Scholar
Erickson, K. B.. Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151 (1970), 263–291.Google Scholar
Feller, W.. An Introduction to Probability Theory and its Applications, II. Wiley, New York, 1966.Google Scholar
Garsia, A. and Lamperti, J.. A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37 (1962/1963), 221–234.Google Scholar
Gouëzel, S.. Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139 (2004), 29–65.Google Scholar
Gouëzel, S.. Vitesse de décorrélation et théorèmes limites pour les applications non uniformément dilatantes. PhD Thesis, Ecole Normale Supérieure, 2004.Google Scholar
Gouëzel, S.. Berry–Esseen theorem and local limit theorem for non uniformly expanding maps. Ann. Inst. Henri Poincaré Probab. Stat. 41 (2005), 997–1024.Google Scholar
Gouëzel, S.. Correlation asymptotics from large deviations in dynamical systems with infinite measure. Colloq. Math. 125 (2011), 193–212.CrossRefGoogle Scholar
Kautzsch, J., Kesseböhmer, M. and Samuel, T.. On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps, Ann. Henri Poincaré, Published online 19 December 2015, doi:10.1007/s00023-015-0451-8.CrossRefGoogle Scholar
Kautzsch, J., Kesseböhmer, M., Samuel, T. and Stratmann, B.. On the asymptotics of the 𝛼-Farey transfer operator. Nonlinearity 28 (2015), 143–166.Google Scholar
Holland, M.. Slowly mixing systems and intermittency maps. Ergod. Th. & Dynam. Sys. 25 (2005), 133–159.Google Scholar
Hu, H. and Vaienti, S.. Polynomial decay for non-uniformly expanding maps. Preprint, 2013, arXiv:1307.0359.Google Scholar
Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19 (1999), 671–685.Google Scholar
Melbourne, I. and Terhesiu, D.. Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. Math. 1 (2012), 61–110.CrossRefGoogle Scholar
Melbourne, I. and Terhesiu, D.. Operator renewal theory and mixing rates for dynamical systems with infinite measure. Corrected version: http://arxiv.org/pdf/1008.4113v3.pdf.Google Scholar
Melbourne, I. and Terhesiu, D.. First and higher order uniform ergodic theorems for dynamical systems with infinite measure. Israel J. Math. 194 (2013), 793–830.Google Scholar
Melbourne, I. and Terhesiu, D.. Erratum to: Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. Math. 202 (2015), 1269–1272.Google Scholar
Melbourne, I. and Terhesiu, D.. Decay of correlation for non-uniformly hyperbolic systems with general return times. Ergod. Th. & Dynam. Sys. 34 (2014), 893–918.Google Scholar
Sarig, O. M.. Subexponential decay of correlations. Invent. Math. 150 (2002), 629–653.Google Scholar
Terhesiu, D.. Error rates in the Darling Kac law. Studia Math. 220(2) (2014), 101–117.CrossRefGoogle Scholar
Terhesiu, D.. Improved mixing rates for infinite measure preserving transformations. Ergod. Th. & Dynam. Sys. 35 (2015), 585–614.Google Scholar
Thaler, M. and Zweimüller, R.. Distributional limit theorems in infinite ergodic theory. Probab. Theory Related Fields 135 (2006), 15–52.Google Scholar
Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.Google Scholar
Zweimüller, R.. Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points. Nonlinearity 11 (1998), 1263–1276.Google Scholar
Zweimüller, R.. Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points. Ergod. Th. & Dynam. Sys. 20 (2000), 1519–1549.Google Scholar