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Coalescence of circle extensions of measure-preserving transformations

Published online by Cambridge University Press:  19 September 2008

Mariusz Lemańczyk ,
Pierre Liardet  and
Jean-Paul Thouvenot
Mariusz Lemańczyk
Affiliation:
Institute of Mathematics, Nicholas Copernicus University, ul. Chopina 12/18, 87–100, Poland
Pierre Liardet
Affiliation:
Université de Provence, U.R. Associée aux CNRS no 225, 3, place V. Hugo, F-13331 Marseille Cedex 3, France
Jean-Paul Thouvenot
Affiliation:
Laboratoire de Probabilités, Université ParisVI, 75252 Paris Cedex 05, France
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Abstract

We prove that for each ergodic automorphism T:(X, ℬ, μ)→(X, ℬ, μ) for which we can find an element S∈C(T) such that the corresponding Z2-action (S, T) on (X, ℬ, μ) is free, there exists a circle valued cocycle φ such that the group extension Tφ is ergodic but is not coalescent. In particular, the existence of such a cocycle is proved for all ergodic rigid automorphisms. As a corollary, in the class of ergodic transformations of [0,1) × [0,1) given by

for each irrational α we find φ such that Tφ is not coalescent. In some special cases the group law of the centralizer is given.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 4 , December 1992 , pp. 769 - 789
Copyright
Copyright © Cambridge University Press 1992

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