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Measure evolution of cellular automata and of finitely anticipative transformations

Published online by Cambridge University Press:  06 October 2015

PIERRE COLLET  and
SERVET MARTÍNEZ
PIERRE COLLET
Affiliation:
Centre de Physique Théorique, CNRS UMR 7644, Ecole Polytechnique F-91128 Palaiseau Cedex, France email collet@cpht.polytechnique.fr
SERVET MARTÍNEZ
Affiliation:
Departamento Ingeniería Matemática and Centro Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS, Casilla 170-3, Correo 3, Santiago, Chile email smartine@dim.uchile.cl
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Abstract

The evolution of cellular automata and of finitely anticipative transformations is studied by using right sets. These are the sets of symbols that are compatible with a past of a position and the respective coordinate of the transformation. Our main result shows, under some suitable conditions, that if the entropy converges to zero then the right sets increase towards the whole alphabet. We discuss these concepts with Wolfram automata.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 1 , February 2017 , pp. 129 - 145
Copyright
© Cambridge University Press, 2015 

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References

Doob, J. L.. Measure Theory (Graduate Texts in Mathematics, 143) . Springer, New York, 1994.Google Scholar
Goles, E., Maass, A. and Martínez, S.. On the limit set of some universal cellular automata. Theoret. Comput. Sci. 110(1) (1993), 53–78.Google Scholar
Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3 (1969), 320–375.Google Scholar
Helvik, T., Lindgren, K. and Nordahl, M. G.. Continuity of information transport in surjective cellular automata. Comm. Math. Phys. 272(1) (2007), 53–74.Google Scholar
Hurd, L.. Formal language characterization of cellular automata limit sets. Complex Systems 1 (1987), 69–80.Google Scholar
Hurd, L.. Recursive cellular automata invariant sets. Complex Systems 4 (1990), 119–129.Google Scholar
Itô, K.. Introduction to Probability Theory. Cambridge University Press, Cambridge, 1984.Google Scholar
Kalikow, S. and McCutcheon, R.. An Outiline of Ergodic Theory (Cambridge Studies in Advanced Mathematics, 122) . Cambridge University Press, Cambridge, 2010.Google Scholar
Maass, A.. On the sofic limit sets of cellular automata. Ergod. Th. & Dynam. Sys. 15 (1995), 663–684.Google Scholar
Rohlin, V. A.. On the fundamental ideas of measure theory. Amer. Math. Soc. Transl. 71 (1952), 1–54.Google Scholar
Wolfram, S.. Computation theory of cellular automata. Comm. Math. Phys. 96 (1984), 15–57.Google Scholar
Zhisong, J.. A complexity analysis of the elementary cellular automaton of rule 122. Appl. Math. J. Chinese Univ. Ser. B 20(3) (2005), 268–276.Google Scholar