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A counterexample to a positive entropy skew product generalization of the Pinsker conjecture

Published online by Cambridge University Press:  19 September 2008

Jonathan King
Jonathan King
Affiliation:
Department of Mathematics, SUNY at Albany, Albany, NY 12222, USA
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Abstract

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The class of k-automorphisms is not contained in a certain class of skew products over a Bernoulli base. The non-identity fibre transformation in the skew is allowed to have positive or even infinite entropy. A difficulty presented by positive entropy is handled via an apparently new property of independent processes (lemma 7.24).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 3 , September 1985 , pp. 379 - 407
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Breiman, L.. Probability. Addison-Wesley, 1968.Google Scholar
[2]Burton, R.. A non-Bernoulli skew product which is loosely Bernoulli. Israel J. Math. 35 (1980), no.4, 339–348.CrossRefGoogle Scholar
[3]Friedman, N.. Introduction to Ergodic Theory. Van Nostrand Reinhold Mathematical Studies #29, 1970.Google Scholar
[4]Meilijson, I.. Mixing properties of a class of skew products. Israel J. Math. 19 (1974), 266–270.Google Scholar
[5]Ornstein, D. S.. A mixing transformation for which Pinsker's conjecture fails. Adv. in Math. 10, No. 1, Feb. 1973, p. 103.Google Scholar
[6]Ornstein, D. S. & Shields, P.. An uncountable family of K-automorphisms. Adv. in Math. 10, No. 1, Feb. 1973, p. 63.CrossRefGoogle Scholar
[7]Ornstein, D., Rudolph, D. & Weiss, B.. Equivalence of measure preserving transformations. Mem. Amer. Math. Soc. 37, No. 262, May 1982.Google Scholar
[8]Rudolph, D.. Two non-isomorphic K-automorphisms with isomorphic squares. Israel J. Math. 23 (1976), 274–287.Google Scholar
[9]Shields, P.. The Theory of Bernoulli Shifts. University of Chicago Press: Chicago and London, 1973.Google Scholar
[10]Shields, P. & Burton, R.. A skew product which is Bernoulli. Mh. Math. 86 (1978), 155–165.CrossRefGoogle Scholar
[11]Thouvenot, Jean-Paul. Quelques propriétés des systécomposent en un produit de deux systèmes dont l'un est un schèma de Bernoulli. Israel J. Math. 21 (1975), 177–207.Google Scholar