Rank, covering number and a simple spectrum
type: Item
instance of: scholarly article
title: Rank, covering number and a simple spectrum
zbMATH Open document ID: 0849.28008
DOI: 10.1007/BF02788822
author: I. Filipowicz
author: J. Kwiatkowski
published in: Journal d'Analyse Mathématique
publication date: 1996-11-10T00:00:00Z
review text: Let \(T\) be an ergodic automorphism on a standard Borel probability space \((X, {\mathcal B}, \mu)\). If \(m\) and \(r\) are positive integers \((m\leq r)\), the question of the existence of such \(T\) having rank \(r\) and maximal spectral multiplicity \(m\) has been studied by several authors (see for example \textit{S. Ferenczi, J. Kwiatkowski} and \textit{C. Mauduit} [J. Anal. Math. 65, 45-75 (1995; Zbl 0833.28010)]). In fact, Kwiatkowski and Lacroix have recently announced that they have solved this problem completely. In this paper, an additional property is required of the transformation \(T\). Roughly speaking, \(T\) has covering number \(b\) if there is a sequence of Rokhlin towers \(X_n\) with \(\mu(X_n)\to b\) as \(n\to \infty\), and such that for every \(A\in {\mathcal B}\), the set \(A\cap X_n\) can be approximated arbitrarily closely by the levels of \(X_n\). If \(b= 1\), then \(T\) has rank 1, and it always is true that if \(r= \text{rank}(T)\), then \(r\cdot b\geq 1\). The idea of positive covering number was introduced by King, Thouvenout and Ferenczi (using the term local rank one) [see \textit{S. Ferenczi}, Ann. Inst. Henri Poincaré, Probab. Stat. 20, 35-51 (1984; Zbl 0535.28010)]. Let \(n\) be an integer. The authors show that given \(b\) and \(r\) satisfying \({1\over n+ 1}\leq b< {1\over n}\) and \(r\geq n+ 1\), then there exists an ergodic \(T\) with \(\text{rank}(T)= r\) and covering number \(b\). It is also shown that there exists \(T\) having a simple spectrum \((m= 1)\), \(\text{rank}(T)= r\) and covering number \(b\) for any positive integer \(r\geq 2\) and any real number \(0< b< 1\) satisfying \(r\cdot b\geq 1\). The examples used are Morse dynamical systems.
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 854441
zbMATH Keywords: ergodic automorphism
zbMATH Keywords: maximal spectral multiplicity
zbMATH Keywords: covering number
zbMATH Keywords: local rank one
zbMATH Keywords: simple spectrum
reviewed by: Geoffrey R. Goodson
MaRDI profile type: Publication
cites work: Q5613310
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full work available at URL: https://doi.org/10.1007/bf02788822
OpenAlex ID: W2020719645
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description: scientific article; zbMATH DE number 854441