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An invariant for rigid rank-1 transformations

Published online by Cambridge University Press:  19 September 2008

Nathaniel Friedman ,
Patrick Gabriel  and
Jonathan King
Nathaniel Friedman
Affiliation:
Department of Mathematics and Statistics, SUNY at Albany, Albany, NY 12222, USA
Patrick Gabriel
Affiliation:
Laboratoire de Probabilités, Université de Dijon, Dijon, France
Jonathan King
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
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Abstract

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Associated to a rigid rank-1 transformation T is a semigroup ℒ(T) of natural numbers, closed under factors. If ℒ(S) ≠ ℒ(T) then S and T cannot be copied isomorphically onto the same space so that they commute. If ℒ(S) ⊅ ℒ(T) then S cannot be a factor of T. For each semigroup L we construct a weak mixing S such that ℒ(S) = L. The S where ℒ(S) = {l}, despite having uncountable commutant, has no roots.

Preceding and preparing for this example are two others: An uncountable abelian group G of weak mixing transformations for which any two (non-identity) members have identical self-joinings of all orders and powers. The second example, to contrast with the rank-1 property that the weak essential commutant must be the trivial group, is of a rank-2 transformation with uncountable weak essential commutant.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 1 , March 1988 , pp. 53 - 72
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

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