Conjugations, joinings, and direct products of locally rank one dynamical systems
type: Item
instance of: scholarly article
title: Conjugations, joinings, and direct products of locally rank one dynamical systems
zbMATH Open document ID: 0996.37002
DOI: 10.1007/BF02463255
author: Valery Ryzhikov
author: Geoffrey R. Goodson
published in: Journal of Dynamical and Control Systems
publication date: 2000-04-13T00:00:00Z
review text: The work is devoted to studying properties of the finite rank automorphism defined on a standard Borel nonatomic probability space. It is shown that if this automorphism is ergodic, has uniform rank \(n\) and \(S\) is an automorphism conjugate to its inverse, then \(S^{2m}=I\) (where \(I\) is the identity automorphism) for some positive integer \(0< m\leq n\). Related results for an ergodic map having positive \(\beta\)-rank with no partial rigidity are given. Further the article is concerned with the question of whether the Cartesian square and its square root can have finite rank. This question is answered in certain cases, in particular, it is proved that in the cases when \(T\) is mixing, \(T\) is not partially rigid, \(T\) is isomorphic to its inverse the Cartesian square and its square root cannot be of finite rank.
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 1431840
zbMATH Keywords: conjugations
zbMATH Keywords: joinings
zbMATH Keywords: finite rank systems
zbMATH Keywords: Cartesian product
zbMATH Keywords: finite rank automorphism
zbMATH Keywords: identity automorphism
zbMATH Keywords: ergodic map
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02463255
OpenAlex ID: W2086717185
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description: scientific article; zbMATH DE number 1431840