Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T03:58:41.970Z Has data issue: false hasContentIssue false
  • English
  • Français

Recurrence in pairs

Published online by Cambridge University Press:  01 August 2008

KAMEL HADDAD  and
WILLIAM OTT
KAMEL HADDAD
Affiliation:
California State University, Bakersfield, Bakersfield, CA 93311, USA (email: [email protected])
WILLIAM OTT
Affiliation:
Courant Institute of Mathematical Sciences, New York, NY 10012, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce and study the notion of weak product recurrence. Two sufficient conditions for this type of recurrence are established. We deduce that any point with a dense orbit in either the full one-sided shift on a finite number of symbols or a mixing subshift of finite type is weakly product recurrent. This observation implies that distality does not follow from weak product recurrence. We have therefore answered, in the negative, a question posed by Auslander and Furstenberg.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 4 , August 2008 , pp. 1135 - 1143
Copyright
Copyright © 2008 Cambridge University Press

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

[1]Auslander, J. and Furstenberg, H.. Product recurrence and distal points. Trans. Amer. Math. Soc. 343(1) (1994), 221–232. MR1170562 (94g:54027).CrossRefGoogle Scholar
[2]Ellis, D. B., Ellis, R. and Nerurkar, M.. The topological dynamics of semigroup actions. Trans. Amer. Math. Soc. 353(4) (2001), 1279–1320 (electronic). MR1806740 (2001m:54041).CrossRefGoogle Scholar
[3]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981, M. B. Porter Lectures. MR603625 (82j:28010).CrossRefGoogle Scholar
[4]van der Waerden, B. L.. Beweis einer Baudetschen Vermutung. Nieuw. Arch. Wiskd. 15 (1927), 212–216.Google Scholar