Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Zero entropy and directional Bernoullicity of a Gaussian $ {\bf Z}\sp 2$-action


Authors: S. Ferenczi and B. Kamiński
Journal: Proc. Amer. Math. Soc. 123 (1995), 3079-3083
MSC: Primary 28D15; Secondary 60G15
DOI: https://doi.org/10.1090/S0002-9939-1995-1209421-9
MathSciNet review: 1209421
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give an example of a Gaussian $ {\mathbb{Z}^2}$-action $ \Phi $ with zero entropy which is weakly mixing, rigid and such that every non-trivial measure-preserving transformation $ {\Phi ^g}$ defined by $ \Phi , g \in {\mathbb{Z}^2}$, is a Bernoulli shift with infinite entropy.


References [Enhancements On Off] (What's this?)

  • [1] J. P. Conze, Entropie d'un groupe abélien de transformations, Z. Wahrsch. Verw. Gebiete 25 (1972), 11-30. MR 0335754 (49:534)
  • [2] I. P. Cornfeld, S. V. Fomin, and Y. G. Sinai, Ergodic theory, Springer-Verlag, Berlin and New York, 1982. MR 832433 (87f:28019)
  • [3] D. Fried, Entropy for smooth abelian actions, Proc. Amer. Math. Soc. 87 (1983), 111-116. MR 677244 (83m:54078)
  • [4] B. Kamiński, The theory of invariant partitions for $ {Z^d}$-actions, Bull. Polish Acad. Sci. Math. 29 (1981), 349-362. MR 640327 (83c:28013)
  • [5] Y. Katznelson and B. Weiss, Commuting measure preserving transformations, Israel J. Math. 12 (1972), 161-173. MR 0316680 (47:5227)
  • [6] A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Sci. SSSR 119 (1958), 861-864. MR 0103254 (21:2035a)
  • [7] W. Parry, Topics in ergodic theory, Cambridge Univ. Press, London and New York, 1981. MR 614142 (83a:28018)
  • [8] M. S. Pinsker, Dynamical systems with completely positive and zero entropy, Dokl. Akad. Nauk SSSR 133 (1960), 1025-1026. MR 0152628 (27:2603)
  • [9] M. Smorodinsky, A partition on a Bernoulli shift which is not weakly Bernoulli, Math. Systems Theory 5 (1971), 201-203. MR 0297971 (45:7023)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28D15, 60G15

Retrieve articles in all journals with MSC: 28D15, 60G15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1209421-9
Keywords: Entropy, Gaussian $ {\mathbb{Z}^2}$-actions, spectral measures
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society