Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T09:44:45.603Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
Get access Rights & Permissions [Opens in a new window]

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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