Spectrum of multidimensional dynamical systems with positive entropy

B. Kamiński; P. Liardet

Studia Mathematica (1994)

  • Volume: 108, Issue: 1, page 77-85
  • ISSN: 0039-3223

Abstract

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Applying methods of harmonic analysis we give a simple proof of the multidimensional version of the Rokhlin-Sinaǐ theorem which states that a Kolmogorov ℤ d -action on a Lebesgue space has a countable Lebesgue spectrum. At the same time we extend this theorem to ℤ ∞ -actions. Next, using its relative version, we extend to ℤ ∞ -actions some other general results connecting spectrum and entropy.

How to cite

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Kamiński, B., and Liardet, P.. "Spectrum of multidimensional dynamical systems with positive entropy." Studia Mathematica 108.1 (1994): 77-85. <http://eudml.org/doc/216042>.

@article{Kamiński1994,
abstract = {Applying methods of harmonic analysis we give a simple proof of the multidimensional version of the Rokhlin-Sinaǐ theorem which states that a Kolmogorov $ℤ^d$-action on a Lebesgue space has a countable Lebesgue spectrum. At the same time we extend this theorem to $ℤ^∞$-actions. Next, using its relative version, we extend to $ℤ^∞$-actions some other general results connecting spectrum and entropy.},
author = {Kamiński, B., Liardet, P.},
journal = {Studia Mathematica},
keywords = {positive entropy; Kolmogorov automorphism; Lebesgue spectrum; -actions; countable Abelian group},
language = {eng},
number = {1},
pages = {77-85},
title = {Spectrum of multidimensional dynamical systems with positive entropy},
url = {http://eudml.org/doc/216042},
volume = {108},
year = {1994},
}

TY - JOUR
AU - Kamiński, B.
AU - Liardet, P.
TI - Spectrum of multidimensional dynamical systems with positive entropy
JO - Studia Mathematica
PY - 1994
VL - 108
IS - 1
SP - 77
EP - 85
AB - Applying methods of harmonic analysis we give a simple proof of the multidimensional version of the Rokhlin-Sinaǐ theorem which states that a Kolmogorov $ℤ^d$-action on a Lebesgue space has a countable Lebesgue spectrum. At the same time we extend this theorem to $ℤ^∞$-actions. Next, using its relative version, we extend to $ℤ^∞$-actions some other general results connecting spectrum and entropy.
LA - eng
KW - positive entropy; Kolmogorov automorphism; Lebesgue spectrum; -actions; countable Abelian group
UR - http://eudml.org/doc/216042
ER -

References

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  1. [G] F. P. Greenleaf, Invariant Means on Topological Groups, Van Nostrand Math. Stud. 16, 1969. Zbl0174.19001
  2. [H] H. Helson, Lectures on Invariant Subspaces, Academic Press, 1964. Zbl0119.11303
  3. [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, 1963. 
  4. [Ka] B. Kamiński, The theory of invariant partitions for ℤ d -actions, Bull. Polish Acad. Sci. Math. 29 (1981), 349-362. 
  5. [Ki1] J. C. Kieffer, A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space, Ann. Probab. 3 (1975), 1031-1037. Zbl0322.60032
  6. [Ki2] J. C. Kieffer, The isomorphism theorem for generalized Bernoulli schemes, in: Studies in Probability and Ergodic Theory, Adv. in Math. Suppl. Stud. 2, Academic Press, 1978, 251-267. 
  7. [Kir] A. A. Kirillov, Dynamical systems, factors and representations of groups, Uspekhi Mat. Nauk 22 (5) (1967), 67-80 (in Russian). 
  8. [MN] V. Mandrekar and M. Nadkarni, Quasi-invariance of analytic measures on compact groups, Bull. Amer. Math. Soc. 73 (1967), 915-920. Zbl0193.10601
  9. [Pa] W. Parry, Topics in Ergodic Theory, Cambridge University Press, 1981. Zbl0449.28016
  10. [Pi] B. S. Pitskel', On informational futures of amenable groups, Dokl. Akad. Nauk SSSR 223 (1975), 1067-1070 (in Russian). 
  11. [RS] V. A. Rokhlin and Ya. G. Sinaǐ, Construction and properties of invariant measurable partitions, ibid. 141 (1961), 1038-1041 (in Russian). 
  12. [Ro] A. Rosenthal, Uniform generators for ergodic finite entropy free actions of amenable groups, Probab. Theory Related Fields 77 (1988), 147-166. Zbl0614.28017
  13. [Ru] W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962. Zbl0107.09603
  14. [T] J. P. Thouvenot, Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l'un est un schéma de Bernoulli, Israel J. Math. 21 (1975), 177-207. Zbl0329.28008

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