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An ergodic transformation with trivial Kakutani centralizer

Published online by Cambridge University Press:  19 September 2008

Adam Fieldsteel  and
Daniel J. Rudolph
Adam Fieldsteel
Affiliation:
Department of Mathematics, Wesleyan University, Middletown, CT 06457, USA
Daniel J. Rudolph
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
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Abstract

Generalizing from the centralizer of a measure-preserving dynamical system (X, ℬ, μ, T), one defines the Kakutani centralizer KC(T) of all ‘even Kakutani factor maps’ ϕ from T to itself. Such a ϕ is a composition φ2ϕ1 of an even Kakutani orbit equivalence ϕ1 and a factor map ϕ2. We construct here an ergodic T acting on a nonatomic Lebesgue space (X,ℬ,μ) with the property that any ϕ ∈ KC(T) is invertible and of the form

All invertible maps of this form are automatically in KC(T) and hence for this T the Kakutani centralizer is as small as possible.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 3 , September 1992 , pp. 459 - 478
Copyright
Copyright © Cambridge University Press 1992

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