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The commutant is the weak closure of the powers, for rank-1 transformations

Published online by Cambridge University Press:  19 September 2008

Jonathan King
Jonathan King
Affiliation:
Department of Mathematics and Statistics, State University of New York at Albany, Albany, NY 12222, USA
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Abstract

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In the class of rank-1 transformations, there is a strong dichotomy. For such a T, the commutant is either irivial, consisting only of the powers of T, or is uncountable. In addition, the commutant semigroup, C(T), is in fact a group. As a consequence, the notion of weak isomorphism between two transformations is equivalent to isomorphism, if at least one of the transformations is rank-1. In § 2, we show that any proper factor of a rank-1 must be rigid. Hence, neither Ornstein's rank-1 mixing nor Chacón's transformation, can be a factor of a rank-1.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 3 , September 1986 , pp. 363 - 384
Copyright
Copyright © Cambridge University Press 1986

References

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