A density theorem for (multiplicity, rank) pairs
type: Item
instance of: scholarly article
title: A density theorem for (multiplicity, rank) pairs
zbMATH Open document ID: 0833.28010
DOI: 10.1007/BF02788765
author: Sébastien Ferenczi
author: J. Kwiatkowski
author: Christian Mauduit
published in: Journal d'Analyse Mathématique
publication date: 1996-03-17T00:00:00Z
review text: Recall that associated with a dynamical system \((X,T, \mu)\) there are certain invariants. The rank \(r\), a conjugacy invariant, is the number of Rokhlin towers necessary to generate the system. The (spectral) multiplicity \(m\), a spectral invariant, is the number of cyclic subspaces necessary to span the \(L^2\)-space for the associated \(L^2\)- operator. It is known that for a dynamical system \(m \leq r\) (Chacon). \textit{M. K. Mentzen} [Bull. Pol. Acad. Sci., Math. 35, 417-424 (1987; Zbl 0675.28006)] conjectured that for each pair \((m,r)\) with \(m \leq r\) (natural numbers), it should be possible to construct a dynamical system with multiplicity \(m\) and rank \(r\), and he showed this is possible for pairs of the form \((1,n)\). Subsequently, this reviewer and \textit{M. Lemańczyk} [Stud. Math. 96, No. 3, 219-230 (1990; Zbl 0711.28007)] showed that \((2,n)\) can be achieved. More recently, other authors have constructed pairs of the form \((n,n)\), \((n,2n)\) and \((p,p - 1)\), \(p\) prime. In this paper the authors construct a new class of examples giving rise to pairs of the form \((d,n)\) for all \(n \geq 3\) and all \(d\) dividing an arithmetic function \(\psi (n)\). In particular, it follows that for fixed \(d\), the set of possible \(n\) is of density one in \({\mathbf N}\).
Mathematics Subject Classification ID: 28D10
zbMATH DE Number: 807400
zbMATH Keywords: spectral multiplicity
zbMATH Keywords: dynamical system
zbMATH Keywords: rank
zbMATH Keywords: conjugacy invariant
zbMATH Keywords: Rokhlin towers
zbMATH Keywords: spectral invariant
reviewed by: Geoffrey R. Goodson
MaRDI profile type: Publication
cites work: Q5613310
cites work: Généralisation du théorème de Ikehara
cites work: A Transformation with Simple Spectrum which is not Rank One
cites work: Systèmes localement de rang un
cites work: Tiling and local rank properties of the Morse sequence
cites work: Rank and spectral multiplicity
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: Strongly mixing g-measures
cites work: Isomorphism of regular Morse dynamical systems
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: Q3309910
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/bf02788765
OpenAlex ID: W2054907494
description: scientific article; zbMATH DE number 807400