Substitution dynamical systems on infinite alphabets

Sébastien Ferenczi[1]

  • [1] Institut de Mathématiques de Luminy CNRS, UPR 9016 163 av. de Luminy 13288 Marseille Cedex 9 (France) and Fédération de Recherche des Unités de Mathématiques de Marseille CNRS - FR 2291

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 7, page 2315-2343
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Ferenczi, Sébastien. "Substitution dynamical systems on infinite alphabets." Annales de l’institut Fourier 56.7 (2006): 2315-2343. <http://eudml.org/doc/10206>.

@article{Ferenczi2006,
abstract = {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.},
affiliation = {Institut de Mathématiques de Luminy CNRS, UPR 9016 163 av. de Luminy 13288 Marseille Cedex 9 (France) and Fédération de Recherche des Unités de Mathématiques de Marseille CNRS - FR 2291},
author = {Ferenczi, Sébastien},
journal = {Annales de l’institut Fourier},
keywords = {Substitutions; dynamical systems; measure-preserving system},
language = {eng},
number = {7},
pages = {2315-2343},
publisher = {Association des Annales de l’institut Fourier},
title = {Substitution dynamical systems on infinite alphabets},
url = {http://eudml.org/doc/10206},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Ferenczi, Sébastien
TI - Substitution dynamical systems on infinite alphabets
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 7
SP - 2315
EP - 2343
AB - 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.
LA - eng
KW - Substitutions; dynamical systems; measure-preserving system
UR - http://eudml.org/doc/10206
ER -

References

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  2. F. DURAND, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998), 89-101 Zbl0895.68087MR1489074
  3. S. FERENCZI, Complexity of sequences and dynamical systems, Discrete Math. 206 (1999), 145-154 Zbl0936.37008MR1665394
  4. E. HOPF, Ergodentheorie, (1937), Springer-Verlag Zbl0185.29001
  5. B. KITCHENS, Symbolic dynamics. One-sided, two-sided and countable state Markov shifts, (1998), Universitext., Springer-Verlag Zbl0892.58020MR1484730
  6. U. KRENGEL, Entropy of conservative transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 7 (1967), 161-181 Zbl0183.19303MR218522
  7. M. LE GONIDEC, Sur la complexité de mots infinis engandrés par des q -automates dénombrables 
  8. C. MAUDUIT, Propriétés arithmétiques des substitutions et automates infinis MR1476736
  9. B. MOSSÉ, Puissances de mots et reconnaissabilité des points fixes d’une substitution, Theoret. Comput. Sci. 99 (1992), 327-334 Zbl0763.68049MR1168468
  10. N. PYTHEAS FOGG, The universal counter-example 
  11. N. PYTHEAS FOGG, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Math. 1794 (2002), Springer-Verlag Zbl1014.11015MR1970385
  12. M. QUEFFÉLEC, Substitution dynamical systems - Spectral analysis, Lecture Notes in Math. 1294 (1987), Springer-Verlag Zbl0642.28013MR924156

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