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Ergodic aspects of cellular automata†

Published online by Cambridge University Press:  19 September 2008

Mike Hurley
Mike Hurley
Affiliation:
Department of Mathematics and Statistics, Case Western Reserve University, Cleveland, Ohio 44106, USA
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Abstract

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This paper contains a study of attractors in cellular automata, particularly the minimal attractors as defined by J. Milnor. Milnor's definition of attractor uses a measure on the state space; the measures that we consider are Bernoulli product measures. Given a Bernoulli measure it is shown that a cellular automaton has at most one minimal attractor; when there is one, it is the omega-limit set of almost all points. Examples are given to show that the minimal attractor can change as the Bernoulli measure is varied. Other examples illustrate the difference between this result and the corresponding result that is obtained by replacing Milnor's definition of attractor by the purely topological definition used by C. Conley. The examples also show that certain invariant sets of cellular automata are less well-behaved than one might hope: for instance the periodic points are not necessarily dense in the nonwandering set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 4 , December 1990 , pp. 671 - 685
Copyright
Copyright © Cambridge University Press 1990

References

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