Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T00:31:08.707Z Has data issue: false hasContentIssue false
  • English
  • Français

On multiple recurrence and other properties of ‘nice’ infinite measure-preserving transformations

Published online by Cambridge University Press:  12 February 2016

JON AARONSON  and
HITOSHI NAKADA
JON AARONSON
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel email [email protected]
HITOSHI NAKADA
Affiliation:
Department of Mathematics, Keio University, Hiyoshi 3-14-1 Kohoku, Yokohama 223, Japan email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss multiple versions of rational ergodicity and rational weak mixing for ‘nice’ transformations, including Markov shifts, certain interval maps and hyperbolic geodesic flows. These properties entail multiple recurrence.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 5 , August 2017 , pp. 1345 - 1368
Copyright
© Cambridge University Press, 2016 

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

Aaronson, J.. Rational ergodicity and a metric invariant for Markov shifts. Israel J. Math. 27(2) (1977), 93–123.CrossRefGoogle Scholar
Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50) . American Mathematical Society, Providence, RI, 1997.Google Scholar
Aaronson, J.. Rational weak mixing in infinite measure spaces. Ergod. Th. & Dynam. Sys. 33(6) (2013), 1611–1643.Google Scholar
Aaronson, J. and Denker, M.. The Poincaré series of ℂ \ ℤ. Ergod. Th. & Dynam. Sys. 19(1) (1999), 1–20.Google Scholar
Aaronson, J. and Denker, M.. Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stoch. Dyn. 1(2) (2001), 193–237.Google Scholar
Aaronson, J., Denker, M., Sarig, O. and Zweimüller, R.. Aperiodicity of cocycles and conditional local limit theorems. Stoch. Dyn. 4(1) (2004), 31–62.Google Scholar
Aaronson, J. and Nakada, H.. Multiple recurrence of Markov shifts and other infinite measure preserving transformations. Israel J. Math. 117 (2000), 285–310.Google Scholar
Aaronson, J. and Sullivan, D.. Rational ergodicity of geodesic flows. Ergod. Th. & Dynam. Sys. 4(2) (1984), 165–178.Google Scholar
Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429–460.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) . revised edition. Springer, Berlin, 2008, With a preface by David Ruelle, Edited by Jean-René Chazottes.Google Scholar
Garsia, A. and Lamperti, J.. A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37 (1962/1963), 221–234.Google Scholar
Guivarc’h, Y. and Hardy, J.. Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. Henri Poincaré Probab. Stat. 24(1) (1988), 73–98.Google Scholar
Hennion, H. and Hervé, L.. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-compactness (Lecture Notes in Mathematics, 1766) . Springer, Berlin, 2001.Google Scholar
Hopf, E.. Ergodentheorie. Number v. 5, no. 2 in Ergebnisse der Mathematik und ihrer Grenzgebiete, 5. Bd. Julius Springer, 1937.Google Scholar
Hopf, E.. Ergodic theory and the geodesic flow on surfaces of constant negative curvature. Bull. Amer. Math. Soc. 77 (1971), 863–877.CrossRefGoogle Scholar
Iwata, Y.. A generalized local limit theorem for mixing semi-flows. Hokkaido Math. J. 37(1) (2008), 215–240; 02.Google Scholar
Krickeberg, K.. Strong mixing properties of Markov chains with infinite invariant measure. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory, Part 2. University of California Press, Berkeley, CA, 1967, pp. 431–446.Google Scholar
Paulin, F., Pollicott, M. and Schapira, B.. Equilibrium states in negative curvature. Preprint, 2012, arXiv:1211.6242.Google Scholar
Rees, M.. Checking ergodicity of some geodesic flows with infinite Gibbs measure. Ergod. Th. & Dynam. Sys. 1(1) (1981), 107–133.Google Scholar
Roblin, T.. Sur l’ergodicité rationnelle et les propriétés ergodiques du flot géodésique dans les variétés hyperboliques. Ergod. Th. & Dynam. Sys. 20(6) (2000), 1785–1819.Google Scholar
Sharp, R.. Closed orbits in homology classes for Anosov flows. Ergod. Th. & Dynam. Sys. 13(2) (1993), 387–408.CrossRefGoogle Scholar
Solomyak, R.. A short proof of ergodicity of Babillot–Ledrappier measures. Proc. Amer. Math. Soc. 129(12) (2001), 3589–3591 (electronic).Google Scholar
Tsuji, M.. Potential Theory in Modern Function Theory. Maruzen Co. Ltd., Tokyo, 1959.Google Scholar
Waddington, S.. Large deviation asymptotics for Anosov flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 13(4) (1996), 445–484.Google Scholar