For mixing transformations \(rank(T^ k)=k\cdot rank(T)\)
type: Item
instance of: scholarly article
title: For mixing transformations \(rank(T^ k)=k\cdot rank(T)\)
zbMATH Open document ID: 0626.47012
DOI: 10.1007/BF02776244
published in: Israel Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: For the class of finite-rank transformations having no partial rigidity (this contains the class of mixing maps) the rank behaves like a logarithm on positive powers in that \(rk(T^ k)=k\cdot rk(T)\). The proof proceeds via a coding argument on generic names. Combined with a group-theoretic argument, this theorem yields a structure theorem for the commutant group of a mixing T: It is a type of twisted product (a ``carry product'') of the integers with a certain finite group called the essential commutant. A consequence of this is that any mixing map with rank no greater than five must have an abelian commutant. [The algebraic argument is done in a more general setting in the article: Joining-rank and the structure of finite rank mixing transformations, which will appear in J. d'Analyse.]
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 4017678
zbMATH Keywords: finite-rank transformations
zbMATH Keywords: partial rigidity
zbMATH Keywords: essential commutant
zbMATH Keywords: mixing map
author: Jonathan L. King
MaRDI profile type: Publication
cites work: The commutant is the weak closure of the powers, for rank-1 transformations
full work available at URL: https://doi.org/10.1007/bf02776244
OpenAlex ID: W2078324071
description: scientific article; zbMATH DE number 4017678