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Substitution dynamical systems on infinite alphabets. (English. French summary) Zbl 1147.37007

Summary: We give a few examples of substitutions on infinite alphabets, and the beginning of a general theory of the associated dynamical systems. In particular, the “drunken man” substitution can be associated to an ergodic infinite measure-preserving system, of Krengel entropy zero, while substitutions of constant length with a positive recurrent infinite matrix correspond to ergodic finite measure preserving systems.

MSC:

37B10 Symbolic dynamics
37A05 Dynamical aspects of measure-preserving transformations
37A40 Nonsingular (and infinite-measure preserving) transformations
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References:

[1] CASSAIGNE, J., Complexité et facteurs spéciaux. Complexity and special factor, Bull. Belg. Math. Soc. Simon Stevin, 4, 4, 1, 67-88 (1997) · Zbl 0921.68065
[2] DURAND, F., A characterization of substitutive sequences using return words, Discrete Math., 179, 89-101 (1998) · Zbl 0895.68087
[3] FERENCZI, S., Complexity of sequences and dynamical systems, Discrete Math., 206, 145-154 (1999) · Zbl 0936.37008
[4] HOPF, E., Ergodentheorie (1937) · Zbl 0185.29001
[5] KITCHENS, B., Symbolic dynamics. One-sided, two-sided and countable state Markov shifts (1998) · Zbl 0892.58020
[6] KRENGEL, U., Entropy of conservative transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 7, 161-181 (1967) · Zbl 0183.19303
[7] LE GONIDEC, M., Sur la complexité de mots infinis engandrés par des \(q\)-automates dénombrables · Zbl 1121.68090
[8] MAUDUIT, C., Propriétés arithmétiques des substitutions et automates infinis · Zbl 1147.11016
[9] MOSSÉ, B., Puissances de mots et reconnaissabilité des points fixes d’une substitution, Theoret. Comput. Sci., 99, 2, 327-334 (1992) · Zbl 0763.68049
[10] PYTHEAS FOGG, N., The universal counter-example
[11] PYTHEAS FOGG, N., Lecture Notes in Math., 1794 (2002) · Zbl 1014.11015
[12] QUEFFÉLEC, M., Lecture Notes in Math., 1294 (1987) · Zbl 0642.28013
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