Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-12T22:02:12.851Z Has data issue: false hasContentIssue false
  • English
  • Français

On rationally ergodic and rationally weakly mixing rank-one transformations

Published online by Cambridge University Press:  26 February 2014

IRVING DAI,
XAVIER GARCIA,
TUDOR PĂDURARIU and
CESAR E. SILVA
IRVING DAI
Affiliation:
Harvard College, University Hall, Cambridge, MA 02138, USA email ifdai@college.harvard.edu
XAVIER GARCIA
Affiliation:
University of Minnesota, Minneapolis, MN 55455-0213, USA email garci363@umn.edu
TUDOR PĂDURARIU
Affiliation:
University of California, Los Angeles, CA 90095-1555, USA email tudor_pad@yahoo.com
CESAR E. SILVA
Affiliation:
Department of Mathematics, Williams College, Williamstown, MA 01267, USA email csilva@williams.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We study the notions of weak rational ergodicity and rational weak mixing as defined by J. Aaronson [Rational ergodicity and a metric invariant for Markov shifts. Israel J. Math. 27(2) (1977), 93–123; Rational weak mixing in infinite measure spaces. Ergod. Th. & Dynam. Sys. 2012, to appear. http://arxiv.org/abs/1105.3541]. We prove that various families of infinite measure-preserving rank-one transformations possess or do not posses these properties, and consider their relation to other notions of mixing in infinite measure.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 4 , June 2015, pp. 1141 - 1164
Copyright
© Cambridge University Press, 2014 

References

Aaronson, J.. Rational ergodicity and a metric invariant for Markov shifts. Israel J. Math. 27(2) (1977), 93–123.CrossRefGoogle Scholar
Aaronson, J.. Rational ergodicity, bounded rational ergodicity and some continuous measures on the circle. Israel J. Math. 33(3–4) (1979), 181–197.CrossRefGoogle Scholar
Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
Aaronson, J.. Conditions for rational weak mixing. 2012. http://arxiv.org/abs/1208.3505.Google Scholar
Aaronson, J.. Rational weak mixing in infinite measure spaces. Ergod. Th. & Dynam. Sys., 2012, to appear. arXiv.org/abs/1105.3541.CrossRefGoogle Scholar
Aaronson, J., Lin, M. and Weiss, B.. Mixing properties of Markov operators and ergodic transformations, and ergodicity of Cartesian products. Israel J. Math. 33(3–4) (1979), 198–224.CrossRefGoogle Scholar
Adams, T., Friedman, N. and Silva, C. E.. Rank-one weak mixing for nonsingular transformations. Israel J. Math. 102 (1997), 269–281.CrossRefGoogle Scholar
Bowles, A., Fidkowski, L., Marinello, A. E. and Silva, C. E.. Double ergodicity of nonsingular transformations and infinite measure-preserving staircase transformations. Illinois J. Math. 45(3) (2001), 999–1019.CrossRefGoogle Scholar
Chacon, R. V.. Weakly mixing transformations which are not strongly mixing. Proc. Amer. Math. Soc. 22 (1969), 559–562.CrossRefGoogle Scholar
Cirilo, P. R., Lima, Y. and Pujals, E.. Law of large numbers for certain cylinder flows. 2011. Preprint, arxiv.org/abs/1108.3519.Google Scholar
Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton Univeristy Press, Princeton, NJ, 1981.CrossRefGoogle Scholar
Gruher, K., Hines, F., Patel, D., Silva, C. E. and Waelder, R.. Power weak mixing does not imply multiple recurrence in infinite measure and other counterexamples. New York J. Math. 9 (2003), 1–22 (electronic).Google Scholar
Hajian, A. B. and Kakutani, S.. Weakly wandering sets and invariant measures. Trans. Amer. Math. Soc. 110 (1964), 136–151.CrossRefGoogle Scholar
Hajian, A. B. and Kakutani, S.. Example of an ergodic measure preserving transformation on an infinite measure space. Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970). Springer, Berlin, 1970, pp. 45–52.CrossRefGoogle Scholar
Silva, C. E.. Invitation to Ergodic Theory (Student Mathematical Library, 42). American Mathematical Society, Providence, RI, 2008.Google Scholar