Multiplicity, rank pairs
type: Item
instance of: scholarly article
title: Multiplicity, rank pairs
zbMATH Open document ID: 0894.28008
DOI: 10.1007/BF02788031
published in: Journal d'Analyse Mathématique
publication date: 1998-08-19T00:00:00Z
review text: This paper is the culmination of the work of numerous authors concerning the problem of M. K. Mentzen. Given a dynamical system \((X,\mu,T)\), the rank \(r(T)\) and spectral multiplicity \(m(T)\) satisfy \(1\leq m(T)\leq r(T)\leq\infty\). \textit{M. K. Mentzen} [Bull. Pol. Acad. Sci., Math. 35, 417-424 (1987; Zbl 0675.28006)] conjectured that any such pair of integers (or \(\infty\)) is attainable by some \textit{ergodic} dynamical system on a non-atomic Lebesgue probability space. Contributions to the solution of this problem have been made by R. V. Chacon, A. del Junco, A. E. Robinson, G. R. Goodson, M. Lemańczyk, M. K. Mentzen, S. Ferenczi, J. Kwiatkowski and C. Mauduit. This paper completes this line of research by answering Mentzen's question affirmatively.
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 28D10
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 1038447
zbMATH Keywords: ergodic dynamical system
zbMATH Keywords: problem of M. K. Mentzen
zbMATH Keywords: rank
zbMATH Keywords: spectral multiplicity
author: J. Kwiatkowski
author: Y. Lacroix
reviewed by: Geoffrey R. Goodson
MaRDI profile type: Publication
cites work: Q5613310
cites work: Rank of Gaussian dynamical systems
cites work: A Transformation with Simple Spectrum which is not Rank One
cites work: Systèmes localement de rang un
cites work: Rank and spectral multiplicity
cites work: A density theorem for (multiplicity, rank) pairs
cites work: On the multiplicity function of ergodic group extensions of rotations
cites work: On the rank of a class of bijective substitutions
cites work: Q4312383
cites work: Strongly mixing g-measures
cites work: Isomorphism of regular Morse dynamical systems
cites work: On the multiplicity function of ergodic group extensions, II
cites work: Spectral properties of G-symbolic Morse shifts
cites work: Toeplitz \(Z_ 2\)-extensions
cites work: The Structure of Generalized Morse Minimal Sets on n Symbols
cites work: Q3829762
cites work: Compact abelian group extensions of discrete dynamical systems
cites work: Ergodic measure preserving transformations with arbitrary finite spectral multiplicities
cites work: Mixing and spectral multiplicity
full work available at URL: https://doi.org/10.1007/bf02788031
OpenAlex ID: W2082965850
description: scientific article; zbMATH DE number 1038447