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Conjugates of Infinite Measure Preserving Transformations

Published online by Cambridge University Press:  20 November 2018

S. Alpern ,
J. R. Choksi  and
V. S. Prasad
S. Alpern
Affiliation:
London School of Economics, London, Great Britain
J. R. Choksi
Affiliation:
McGill University, Montreal, Quebec
V. S. Prasad
Affiliation:
York University, North York, Ontario
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In this paper we consider a question concerning the conjugacy class of an arbitrary ergodic automorphism σ of a sigma finite Lebesgue space (X, , μ) (i.e., a is a ju-preserving bimeasurable bijection of (X, , μ). Specifically we prove

THEOREM 1. Let τ, σ be any pair of ergodic automorphisms of an infinite sigma finite Lebesgue space (X, , μ). Let F be any measurable set such that

Then there is some conjugate σ' of σ such that σ'(x) = τ(x) for μ-almost every x in F.

The requirement that F ∪ τF has a complement of infinite measure is, for example, satisfied when F has finite measure, and in that case, the theorem was proved by Choksi and Kakutani ([7], Theorem 6).

Conjugacy theorems of this nature have proved to be very useful in proving approximation results in ergodic theory. These conjugacy results all assert the denseness of the conjugacy class of an ergodic (or antiperiodic) automorphism in various topologies and subspaces.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 3 , 01 June 1988 , pp. 742 - 749
Copyright
Copyright © Canadian Mathematical Society 1988

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