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Two facts concerning the transformations which satisfy the weak Pinsker property

Published online by Cambridge University Press:  01 April 2008

J.-P. THOUVENOT
J.-P. THOUVENOT*
Affiliation:
Laboratoire de Probabilites et Modeles Aleatoires, UMR 7599, Universites Paris 6 et Paris 7, Boite Courrier 188, 4, Place Jussieu, 75252 Paris Cedex 05, France (email: kalikow@ccr.jussieu.fr)
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Abstract

We show that every ergodic, finite entropy transformation which satisfies the weak Pinsker property possesses a finite generator whose two-sided tail field is exactly the Pinsker algebra. This is proved by exhibiting a generator endowed with a block structure quite analogous to the one appearing in the construction of the Ornstein–Shields examples of non Bernoulli K-automorphisms. We also show that, given two transformations T1 and T2 in the previous class (i.e. satisfying the weak Pinsker property), and a Bernoulli shift B, if T1×B is isomorphic to T2×B, then T1 is isomorphic to T2. That is, one can ‘factor out’ Bernoulli shifts.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008 , pp. 689 - 695
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Ornstein, D. S. and Shields, P. C.. An uncountable family of K-automorphisms. Adv. Math. 10 (1973), 103–120.CrossRefGoogle Scholar
[2]Ornstein, D. S. and Weiss, B.. Every transformation is bilaterally deterministic. Israel J. Math. 21(2–3) (1975), 154–158.CrossRefGoogle Scholar
[3]Rahe, M.. Relatively finitely determined implies relatively very weak Bernoulli. Canad. J. Math. 30(3) (1978), 531–548.CrossRefGoogle Scholar
[4]Thouvenot, J.-P.. Quelque propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schema de Bernoulli. Israel J. Math. 21(2–3) (1975), 177–203.CrossRefGoogle Scholar
[5]Thouvenot, J.-P.. Remarques sur les systèmes dynamiques donnés avec plusieurs facteurs. Israel J. Math. 21(2–3) (1975), 215–232.CrossRefGoogle Scholar
[6]Thouvenot, J.-P.. On the stability of the weak Pinsker property. Israel J. Math. 27(2) (1977), 150–162.CrossRefGoogle Scholar
[7]Vershik, A.. Towards the definition of metric hyperbolicity. Moscow Math. J. 5(3) (2005), 721–737.CrossRefGoogle Scholar