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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.

Information

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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References

REFERENCES

[delJ, R]del Junco, A. & Rudolph, D. J.. Kakutani equivalence of ergodic Zn actions. Ergod. Th. & Dynam. Sys. 4 (1984), 89–104.CrossRefGoogle Scholar
[delJ, F, R]del Junco, A., Fieldsteel, A. & Rudolph, D. J., α-equivalence of ergodic transformations. In preparation.Google Scholar
[F, N]Feldman, J. & Nadler, D.. Reparametrization of N-flows of zero entropy. Trans. Amer. Math. Soc. 256 (1979), 289–304.Google Scholar
[F, F]Fieldsteel, A. & Friedman, N. A.. Restricted orbit changes of ergodic Zd-actions to achieve mixing and completely positive entropy. Ergod. Th. & Dynam. Sys. 6 (1986), 505–528.CrossRefGoogle Scholar
[L]Lemanczyk, M.. Weakly isomorphic transformations that are not isomorphic. Prob. Th. Rel. Fields 78 (1988), 491–507.CrossRefGoogle Scholar
[L, L]Lemanczyk, M. & Liardet, P.. Coalescence of Anzai skew products. Preprint.Google Scholar
[K]Kakutani, S.. Induced measure-preserving transformations. Proc. Imp. Acad. Tokyo 19 (1943), 635–641.Google Scholar
[N]Newton, D.. Coalescence and spectrum of automorphisms of a Lebesgue space. Z. Wahr. verw. Geb. 19 (1971), 117–122.CrossRefGoogle Scholar
[O, R, W]Ornstein, D. S., Rudolph, D. J. & Weiss, B.. Equivalence of measure-preserving transformations. Mem. Amer. Math. Soc. 37, no. 262 (1982).Google Scholar
[R1]Rudolph, D. J.. Smooth orbit equivalence of ergodic ℝd actions, d ≥ 2. Trans. Amer. Math. Soc. 253 (1979), 291–302.Google Scholar
[R2]Rudolph, D. J.. Inner and barely linear time changes of ergodic ℝk actions. Contemp. Math. 26 (1984), 351–371.CrossRefGoogle Scholar
[R3]Rudolph, D. J.. Restricted orbit equivalence. Mem. Amer. Math. Soc. 54 no. 323 (1985).Google Scholar
[R4]Rudolph, D. J.. Classifying the isometric extension of a Bernoulli shift. J. a'Analyse Math. 34 (1978), 36–59.CrossRefGoogle Scholar