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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: khaddad@cs.csubak.edu)
WILLIAM OTT
Affiliation:
Courant Institute of Mathematical Sciences, New York, NY 10012, USA (email: ott@cims.nyu.edu)
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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.

Information

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

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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