Abstract
In this article we deal with two of the numerous instances in which automata play a part in Topological or Measurable Dynamics. The first is preservation of normality by transducers; here we give a detailed account of the main proof in [2], that of normality preservation under multiplication by rationals. The second instance is an introduction to the dynamical properties of onto cellular automata—since there are but a few known results about them, we mainly give definitions, point out some elementary properties and ask questions. These two applications of automata in the field of Dynamics, though very different in spirit, are strongly linked, because cellular automata are a particular class of transducers, because entropy and other, mainly topological, notions from Dynamical Systems play an important part in both, and finally because one of the underlying aims is to study the transformations of the interval associated to some of these automata.
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References
Blanchard, F., Non Literal Transducers and Some Problems of Normality, J. Th. Nombres Bordeaux,to appear.
Blanchard, F., Dumont, J.-M., Thomas, A., Generic Sequences, Transducers and Multiplication of Normal Numbers, Israel J. Math., 80, 257–287 (1992).
Blanchard, F., Host, B., Maass, A., Représentation par Automates de Fonctions Continues du Tore, preprint (1993).
Botelho, F., Garzon, M., On Dynamical Properties of Neural Networks, Complex Systems 5, 401–413 (1991).
Broglio, A., Liardet, P., Prediction with Automata. Symbolic Dynamics and its Applications, P. Walters (ed.), Contemporary Math. 135, AMS, Providence (1985).
Coven, E.M., Topological Entropy of Block Maps, Proc. Amer. Math. Soc. 78, 590–594 (1980).
Denker, M., Grillenberger, C., Sigmund, K., Ergodic Theory on Compact Spaces, Lecture Notes in Math. 527, Springer, Berlin (1976).
Fischer, R., Sofic Systems and Graphs, Monats. Math. 80, 179–186 (1975).
Furstenberg, H., Disjointness in Ergodic Theory, Minimal Sets, and a Problem of Diophantine Approximation, Math. Systems Theory 1, 1–49 (1967).
Hedlund, G.A., Endomorphisms and Automorphisms of the Shift Dynamical System, Math. Systems Theory 3, 320–375 (1969).
Hurd, L. P., Kari, J., Culik, K., The Topological Entropy of Cellular Automata is Undecidable, Ergodic Th. Dynam. Sys 12, 255–265 (1992).
Goles, E., Martinez, S., Automata Networks, Dynamical Systems and Statistical Physics, Kluwer Acadamic Pub., Mathematics and its Applications, (1992).
Johnson, A.S., Measures on the Circle Invariant under Multiplication by a Nonlacunary Subsemigroup of the Integers, preprint (1991).
Kamae, T., Weiss, B., Normal Numbers and Selection Rules, Israel J. Math. 21, 101–110 (1975).
Kari, J., Decision Problems Concerning Cellular Automata, Thesis, University of Turku, Finland (1990).
Lind, D.A., Application of Ergodic Theory and Sofic Systems to Cellular Automata, Physica 10 D, 36–44 (1984).
Lind, D.A., Entropies of Automorphisms of a Topological Markov Shift, Proc. Amer. Math. Soc. 99, 589–595 (1987).
Maass, A., On Sofic Limit Sets of Cellular Automata, preprint (1992).
Parry, W., Topics in Ergodic Theory, Cambridge University Press, London (1981) .
Thomas, A., Suites Normales et Transducteurs, preprint (1992).
Walters, P., An Introduction to Ergodic Theory, Graduate Texts in Math. 79, Springer, Berlin (1982).
Wolfram, S., Theory and Applications of Cellular Automata, World Scientific, Singapore (1986).
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© 1994 Springer Science+Business Media Dordrecht
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Blanchard, F. (1994). Cellular Automata and Transducers. A Topological View. In: Goles, E., Martínez, S. (eds) Cellular Automata, Dynamical Systems and Neural Networks. Mathematics and Its Applications, vol 282. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1005-3_1
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DOI: https://doi.org/10.1007/978-94-017-1005-3_1
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