Generic points in the cartesian powers of the Morse dynamical system

Emmanuel Lesigne; Anthony Quas; Máté Wierdl

Bulletin de la Société Mathématique de France (2003)

  • Volume: 131, Issue: 3, page 435-464
  • ISSN: 0037-9484

Abstract

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The symbolic dynamical system associated with the Morse sequence is strictly ergodic. We describe some topological and metrical properties of the Cartesian powers of this system, and some of its other self-joinings. Among other things, we show that non generic points appear in the fourth power of the system, but not in lower powers. We exhibit various examples and counterexamples related to the property of weak disjointness of measure preserving dynamical systems.

How to cite

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Lesigne, Emmanuel, Quas, Anthony, and Wierdl, Máté. "Generic points in the cartesian powers of the Morse dynamical system." Bulletin de la Société Mathématique de France 131.3 (2003): 435-464. <http://eudml.org/doc/272360>.

@article{Lesigne2003,
abstract = {The symbolic dynamical system associated with the Morse sequence is strictly ergodic. We describe some topological and metrical properties of the Cartesian powers of this system, and some of its other self-joinings. Among other things, we show that non generic points appear in the fourth power of the system, but not in lower powers. We exhibit various examples and counterexamples related to the property of weak disjointness of measure preserving dynamical systems.},
author = {Lesigne, Emmanuel, Quas, Anthony, Wierdl, Máté},
journal = {Bulletin de la Société Mathématique de France},
keywords = {topological dynamics; ergodic theory; symbolic dynamical systems; Morse sequence; odometer; joinings; generic points; weak disjointness},
language = {eng},
number = {3},
pages = {435-464},
publisher = {Société mathématique de France},
title = {Generic points in the cartesian powers of the Morse dynamical system},
url = {http://eudml.org/doc/272360},
volume = {131},
year = {2003},
}

TY - JOUR
AU - Lesigne, Emmanuel
AU - Quas, Anthony
AU - Wierdl, Máté
TI - Generic points in the cartesian powers of the Morse dynamical system
JO - Bulletin de la Société Mathématique de France
PY - 2003
PB - Société mathématique de France
VL - 131
IS - 3
SP - 435
EP - 464
AB - The symbolic dynamical system associated with the Morse sequence is strictly ergodic. We describe some topological and metrical properties of the Cartesian powers of this system, and some of its other self-joinings. Among other things, we show that non generic points appear in the fourth power of the system, but not in lower powers. We exhibit various examples and counterexamples related to the property of weak disjointness of measure preserving dynamical systems.
LA - eng
KW - topological dynamics; ergodic theory; symbolic dynamical systems; Morse sequence; odometer; joinings; generic points; weak disjointness
UR - http://eudml.org/doc/272360
ER -

References

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  1. [1] H. Furstenberg – « Strict ergodicity and transformations of the torus », Amer. J. Math83 (1961), p. 573–601. Zbl0178.38404MR133429
  2. [2] E. Lesigne, C. Mauduit & B. Mossé – « Le théorème ergodique le long d’une suite q -multiplicative », Comp. Math.93 (1994), p. 49–79. Zbl0818.28006MR1286798
  3. [3] E. Lesigne, A. Quas, T. de la Rue & B. Rittaud – « Weak disjointness in ergodic theory », Proceedings of the Conference on Ergodic Theory and Dynamical Systems, Toruń 2000, Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Toruń, Poland, 2001. 
  4. [4] E. Lesigne, B. Rittaud & T. de la Rue – « Weak disjointness of measure preserving dynamical systems », to appear in Ergodic Theory & Dynamycal Systems. Available in PDF format as http://www.csi.hu/mw/Weak-Disj.TEB.2.pdf and in DVI format as http://www.csi.hu/mw/Weak-Disj.TEB.2.dvi. Zbl1083.37003
  5. [5] C. Mauduit – « Substitutions et ensembles normaux », Habilitation à diriger des recherches, Université Aix-Marseille II, 1989. 
  6. [6] M. Morse – « Recurrent geodesics on a surface of negative curvature », Trans. Amer. Math. Soc.22 (1921), p. 84–100. Zbl48.0786.06MR1501161JFM48.0786.06
  7. [7] E. Prouhet – « Mémoire sur quelques relations entre les puissances des nombres », C. R. Acad. Sci. Paris 33 (1851), p. 225. 
  8. [8] A. Thue – « Uber unendliche Zeichenreihen (1906), uber die gegenseitige lage gleicher teile gewisser zeichenreihen (1912) », Selected mathematical papers of Axel Thue, Universitetsforlaget, 1977. JFM39.0283.01

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