On ergodic transformations with homogeneous spectrum
type: Item
instance of: scholarly article
title: On ergodic transformations with homogeneous spectrum
zbMATH Open document ID: 0943.37005
DOI: 10.1023/A:1021701019156
author: Oleg N. Ageev
published in: Journal of Dynamical and Control Systems
publication date: 2000-04-13T00:00:00Z
review text: An old question of Rokhlin asks whether there exists an ergodic automorphism having a homogeneous spectrum of multiplicity two (i.e., the multiplicity function of the induced unitary operator takes the unique value 2, almost everywhere with respect to the maximal spectral type). It was recently shown that if \(A\) is any subset of \(\mathbb{Z}^+\cup \{\infty\}\) with \(1\in A\), then there is an ergodic automorphism whose multiplicity function has range \(A\) [see \textit{J. Kwiatkowski} and \textit{M. Lemańczyk}, Stud. Math. 116, 207-215 (1995; Zbl 0857.28012)], but the more general question is still open (for example, is there an ergodic automorphism having \(A= \{n\}\), \(n>2 \)?) The question of Rokhlin is answered affirmatively, as the author shows that for a generic set of ergodic transformations \(T\), \(T\times T\) has a homogeneous spectrum of multiplicity two. It should be noted that \textit{V. V. Ryzhikov} has simultaneously and independently solved this problem in a similar way [J. Dyn. Control Syst. 5, 145-148 (1999)]. See also the survey article of the reviewer [J. Dyn. Control Syst. 5, 173-226 (1999)], which gives a detailed look at the solution of Ryzhikov and also surveys recent results concerning the spectral theory of ergodic dynamical systems. The author is also able to answer affirmatively a conjecture of A. Katok concerning the range of the multiplicity function of Cartesian powers \(T\times T\times T\) (and higher order powers), for the generic transformation \(T\).
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 1431877
zbMATH Keywords: homogeneous spectrum
zbMATH Keywords: Cartesian product
zbMATH Keywords: ergodic automorphism
zbMATH Keywords: homogeneous spectrum of multiplicity two
zbMATH Keywords: range of the multiplicity function of Cartesian powers
reviewed by: Geoffrey R. Goodson
MaRDI profile type: Publication
full work available at URL: a:1021701019156
OpenAlex ID: W113909926
description: scientific article; zbMATH DE number 1431877