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Spring Topology and Dynamics Conference 2006
March 23-25, 2006
University of North Carolina at Greensboro
Greensboro, NC, USA |
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Organizers
Gregory Bell, Alexander Chigogidze, Paul Duvall, Jan Rychtar, Jerry Vaughan
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Prevalence of Odometers in Cellular Automata
by
Ethan M. Coven
Wesleyan University
Coauthors: M. Pivato and R. Yassawi (Trent University, Canada)
Abstract
We consider left permutative cellular automata with no memory and positive anticipation,
defined on all doubly infinite sequences with entries from a finite alphabet. For some of
these automata, including all those defined on 2-letter alphabets which are "linear in the
first variable, " there is a dense set of points such that the automaton restricted to the
closure of the positive semi-orbit of any of these points is topologically conjugate to the n-adic odometer (n = size of alphabet), the "+1" map on the n-adic integers. For the rest of these automata, there is a dense set of points such that the automaton restricted to the closure of the positive semi-orbit of any of these points is topologically conjugate to a generalized odometer, the "+1" map on a countable product of finite groups, addition "with carrying." We identify the (generalized) odometer is some cases.
For each fixed point z of the automaton, the set of points of the form y.z, y arbitrary,
where the dot lies between places 0 and 1, such that the automaton restricted to the closure
of the positive semi-orbit of any of these points is topologically conjugate to an odometer or to a generalized odometer is a dense G-delta subset of {y.z: y arbitrary}, equal to the set of points in {y.z: y arbitrary} with infinite orbits.
Date received: February 6, 2006
Copyright © 2006 by the author(s).
The author(s) of this document and the organizers of the conference
have granted their consent to include this abstract in
Atlas Conferences Inc.
Document # caqs-29.