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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Prevalence of odometers in cellular automata


Authors: Ethan M. Coven, Marcus Pivato and Reem Yassawi
Journal: Proc. Amer. Math. Soc. 135 (2007), 815-821
MSC (2000): Primary 37B10, 37B15
Published electronically: September 15, 2006
MathSciNet review: 2262877
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Abstract: We consider left permutive cellular automata $ \Phi$ with no memory and positive anticipation, defined on the space of all doubly infinite sequences with entries from a finite alphabet. For each such automaton that is not one-to-one, there is a dense set of points $ x$ such that $ \Phi : \operatorname{cl} \{\Phi^n(x) : n \ge 0\} \to \operatorname{cl} \{\Phi^n(x) : n \ge 0\}$ is topologically conjugate to an odometer, the ``$ +1$'' map on the countable product of finite cyclic groups. This set is a dense $ G_\delta$ subset of an appropriate subspace. We identify the odometer in several cases.


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Additional Information

Ethan M. Coven
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06457-0128
Email: ecoven@wesleyan.edu

Marcus Pivato
Affiliation: Department of Mathematics, Trent University, Peterborough, Ontario, Canada K9L 1Z8
Email: pivato@xaravve.trentu.ca

Reem Yassawi
Affiliation: Department of Mathematics, Trent University, Peterborough, Ontario, Canada K9L 1Z8
Email: ryassawi@trentu.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08754-5
PII: S 0002-9939(06)08754-5
Keywords: Odometer, cellular automaton
Received by editor(s): October 15, 2005
Published electronically: September 15, 2006
Additional Notes: This work was done in Spring 2005 while the second and third authors were van Vleck Visiting Professors of Mathematics at Wesleyan University. The first author wishes to thank the lovely summer weather on Cape Cod for delaying the submission of this article.
Communicated by: Michael Handel
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.