Author:
Friedman Nathaniel,Gabriel Patrick,King Jonathan
Abstract
AbstractAssociated to a rigid rank-1 transformationTis a semigroup ℒ(T) of natural numbers, closed under factors. If ℒ(S) ≠ ℒ(T) thenSandTcannot be copied isomorphically onto the same space so that they commute. If ℒ(S) ⊅ ℒ(T) thenScannot be a factor ofT. For each semigroupLwe construct a weak mixingSsuch that ℒ(S) =L. TheSwhere ℒ(S) = {l}, despite having uncountable commutant, has no roots.Preceding and preparing for this example are two others: An uncountable abelian groupGof 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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,General Mathematics
Reference4 articles.
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