A lower bound on the rank of mixing extensions
type: Item
instance of: scholarly article
title: A lower bound on the rank of mixing extensions
zbMATH Open document ID: 0647.28012
DOI: 10.1007/BF02774146
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let T be a measure-preserving transformation of a probability space (X,\(\mu)\). A partition \({\mathbb{P}}\) is a finite list of disjoint measurable sets whose union is all of X. For two partitions, say that \({\mathbb{P}}\epsilon\)-refines \({\mathbb{Q}}\) if for each member \(A\in {\mathbb{Q}}\) there exists a set B, a union of members of \({\mathbb{P}}\), such that \(\mu (A\Delta B)<\epsilon.\) Suppose a natural number r has the following property: Given \(\epsilon\) and any partition \({\mathbb{Q}}\), there exists r disjoint Rohlin stacks of a common height such that, letting \({\mathbb{P}}\) be the partition whose members are the levels of the stacks (and the complement of their union), \({\mathbb{P}}\epsilon\)-refines \({\mathbb{Q}}\). The rank of T is the infimum of all such numbers r. The result: If transformation R is a k-point extension of transformation T then \[ rank(R)\geq k\quad \cdot \quad rank(T) \] whenever T is mixing. Indeed, this remains true with the weaker requirement only that T have zero-rigidity. This is a second application of a technique developed in Isr. J. Math. 56, 102-122 (1986; Zbl 0626.47012)]. The fundamental lemma is proven there.
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 4055966
zbMATH Keywords: measure-preserving transformation of a probability space
zbMATH Keywords: partition
zbMATH Keywords: Rohlin stacks
zbMATH Keywords: rigidity
author: Jonathan L. King
MaRDI profile type: Publication
cites work: For mixing transformations \(rank(T^ k)=k\cdot rank(T)\)
cites work: Joining-rank and the structure of finite rank mixing transformations
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/bf02774146
OpenAlex ID: W2057279171
description: scientific article; zbMATH DE number 4055966