Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-06T05:09:47.857Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence and divergence of ergodic averages

Published online by Cambridge University Press:  19 September 2008

Roger L. Jones  and
Mate Wierdl
Roger L. Jones
Affiliation:
Department of Mathematics, DePaul University, 2219 N. Kenmore, Chicago, IL 60614, USA
Mate Wierdl
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60201, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider almost everywhere convergence and divergence properties of various ergodic averages. A general method is given which can be used to construct averages for which a.e. convergence fails, and to show divergence (and in some cases ‘strong sweeping out’) for large classes of ergodic averages. We also show that there are sequences with the gaps between successive terms converging to zero, but such that the Cesaro averages obtained by sampling a flow along these sequences of times converge a.e. for all f∈L1(X).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 3 , September 1994 , pp. 515 - 535
Copyright
Copyright © Cambridge University Press 1994

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

REFERENCES

[1]Akcoglu, M., Bellow, A., Junco, A. del and Jones, R.. Divergence of averages obtained by sampling a flow. Proc. Amer. Math. Soc. 118 (1993), 499–505.CrossRefGoogle Scholar
[2]Akcoglu, M., Bellow, A., Jones, R., Losert, V., Reinholt-Larssen, K. and Wierdl, M.. The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers and related matters. Ergod. Th. & Dynam. Sys. To appear.Google Scholar
[3]Akcoglu, M. and Junco, A. del. Convergence of averages of point transformations. Proc. Amer. Math. Soc. 49 (1975), 265–266.Google Scholar
[4]Akcoglu, M., Junco, A. del and Lee, W.. A solution to a problem of A. Bellow. Almost Everywhere Convergence II. Bellow, A. and Jones, R., eds. Academic Press: New York, 1991.Google Scholar
[5]Bellow, A.. On Bad Universal Sequences in Ergodic Theory (II), Measure Theory and its Applications, Springer Lecture Notes in Mathematics 1033. Springer: Berlin—New York, 1983, pp. 74–78.Google Scholar
[6]Bellow, A.. Perturbation of a sequence. Adv. Math. 78 (1989), 131–139.CrossRefGoogle Scholar
[7]Bellow, A., Jones, R. and Rosenblatt, J.. Almost everywhere convergence of weighted averages. Math. Ann. 293 (1992), 399–426.CrossRefGoogle Scholar
[8]Bellow, A., Jones, R. and Rosenblatt, J.. Convergence for moving averages. Ergod. Th. & Dynam. Sys. 10 (1990), 43–62.CrossRefGoogle Scholar
[9]Bergelson, V., Boshernitzan, M. and Bourgain, J.. Some results on non-linear recurrence. J. D'Analyse Math. 62 (1994), 29–46.CrossRefGoogle Scholar
[10]Besicovitch, A.. On the linear independence of fractional powers of integers. J. London Math. Soc. 15 (1940), 3–6.CrossRefGoogle Scholar
[11]Bourgain, J.. Almost sure convergence and bounded entropy. Israel J. Math. 73 (1988) 79–97.CrossRefGoogle Scholar
[12]Boshernitzan, M.. Personnel communication.Google Scholar
[13]Junco, A. del and Rosenblatt, J.. Counterexamples in ergodic theory and number theory. Math. Ann. 247 (1984), 185–197.Google Scholar
[14]Emerson, W.. The pointwise ergodic theorem for amenable groups. Amer. J. Math. 96 (1974), 472–487.CrossRefGoogle Scholar
[15]Jessen, B.. On the approximation of the Lebesgue integrals by Riemann sums. Ann. Math. 35(2) (1934), 248–251.CrossRefGoogle Scholar
[16]Jones, R., Olsen, J. and Wierdl, M.. Subsequence ergodic theorems for Lp contractions, Trans. Amer. Math. Soc. 331 (1992), 837–850.Google Scholar
[17]Krengel, U.. On the individual ergodic theorem for subsequences. Ann. Math. Stat. 42 (1971), 1091–1095.CrossRefGoogle Scholar
[18]Lind, D.. Locally compact measure preserving flows. Adv. Math. 15 (1975), 175–193.CrossRefGoogle Scholar
[19]Niven, I. and Zuckerman, H.. An Introduction to the Theory of Numbers, 2ed., Wiley: Chichester, 1960, p. 249.Google Scholar
[20]Reinhold-Larsson, K.. Phd Thesis. Ohio State University, 1991.Google Scholar
[21]Rosenblatt, J.. Universally bad sequences in ergodic theory. In Almost Everywhere Convergence II. Bellow, A. and Jones, R., eds. Academic Press: New York, 1991.Google Scholar
[22]Schwartz, M.. Polynomially moving ergodic averages. Proc. Amer. Math. Soc. 103 (1988), 252–254.CrossRefGoogle Scholar
[23]Tempelman, A.. Ergodic theorems for general dynamical systems. Trudy Moskov. Mat. Obsc. 26.Google Scholar
(Engl. Trans. Moscow Math. Soc. 26 (1972), 94–132.)Google Scholar