Skip to main content
×
×
Home

Improved mixing rates for infinite measure-preserving systems

  • DALIA TERHESIU (a1)
Abstract

In this work, we introduce a new technique for operator renewal sequences associated with dynamical systems preserving an infinite measure that improves the results on mixing rates obtained by Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. Math. 1 (2012), 61–110]. Also, this technique allows us to offer a very simple proof of the key result of Melbourne and Terhesiu that provides first-order asymptotics of operator renewal sequences associated with dynamical systems with infinite measure. Moreover, combining techniques used in this work with techniques used by Melbourne and Terhesiu, we obtain first-order asymptotics of operator renewal sequences under some relaxed assumption on the first return map.

Copyright
References
Hide All
[1]Aaronson, J.. Random $f$-expansions. Ann. Probab. 14 (1986), 10371057.
[2]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.
[3]Aaronson, J. and Denker, M.. Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stoch. Dyn. 1 (2001), 193237.
[4]Aaronson, J., Denker, M. and Urbański, M.. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337 (1993), 495548.
[5]Bingham, N. H., Goldie, C. M. and Teugels, J. L.. Regular Variation (Encyclopedia of Mathematics and its Applications, 27). Cambridge University Press, Cambridge, 1987.
[6]Darling, D. A. and Kac, M.. On occupation times for Markoff processes. Trans. Amer. Math. Soc. 84 (1957), 444458.
[7]DeTemple, D. W.. A quicker convergence to the Euler constant. Amer. Math. Monthly 100 (1993), 468470.
[8]Erickson, K. B.. Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151 (1970), 263291.
[9]Feller, W.. An Introduction to Probability Theory and its Applications, II. Wiley, New York, 1966.
[10]Garsia, A. and Lamperti, J.. A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37 (1962/1963), 221234.
[11]Gouëzel, S.. Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139 (2004), 2965.
[12]Gouëzel, S.. Characterization of weak convergence of Birkhoff sums for Gibbs–Markov maps. Israel J. Math. 180 (2010), 141.
[13]Gouëzel, S.. Correlation asymptotics from large deviations in dynamical systems with infinite measure. Colloq. Math. 125 (2011), 193212.
[14]Gouëzel, S.. Berry–Esseen theorem and local limit theorem for non uniformly expanding maps. Ann. Inst. Henri Poincaré Probab. Stat. 41 (2005), 9971024.
[15]Holland, M.. Slowly mixing systems and intermittency maps. Ergod. Th. & Dynam. Sys. 25 (2005), 133159.
[16]Hu, H. and Vaienti, S.. Absolutely continuous invariant measures for non-uniformly hyperbolic maps. Ergod. Th. & Dynam. Sys. 29 (2009), 11851215.
[17]Kato, T.. Perturbation Theory of Linear Operators (Grundlehren der Mathematischen Wissenschaften, 132). Springer, New York, 1976.
[18]Lamperti, J.. Some limit theorems for stochastic processes. J. Math. Mech. 7 (1958), 433448.
[19]Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19 (1999), 671685.
[20]Melbourne, I. and Terhesiu, D.. Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. Math. 1 (2012), 61110.
[21]Melbourne, I. and Terhesiu, D.. First and higher order uniform ergodic theorems for dynamical systems with infinite measure. Israel J. Math. 194 (2013), 793830.
[22]Pomeau, Y. and Manneville, P.. Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74 (1980), 189197.
[23]Sarig, O. M.. Subexponential decay of correlations. Invent. Math. 150 (2002), 629653.
[24]Thaler, M.. A limit theorem for the Perron–Frobenius operator of transformations on $[0, 1] $ with indifferent fixed points. Israel J. Math. 91 (1995), 111127.
[25]Thaler, M.. The Dynkin–Lamperti arc-sine laws for measure preserving transformations. Trans. Amer. Math. Soc. 350 (1998), 45934607.
[26]Thaler, M. and Zweimüller, R.. Distributional limit theorems in infinite ergodic theory. Probab. Theory Related Fields 135 (2006), 1552.
[27]Zweimüller, R.. Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points. Ergod. Th. & Dynam. Sys. 20 (2000), 15191549.
[28]Zygmund, A.. Trigonometric Series. 2nd edn. Vol. 1, Cambridge University Press, Cambridge, 1968.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 12 *
Loading metrics...

Abstract views

Total abstract views: 113 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 12th June 2018. This data will be updated every 24 hours.