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A Measure Preserving Transformation whose Spectrum has Lebesgue Component of Multiplicity Two

A Measure Preserving Transformation whose Spectrum has Lebesgue Component of Multiplicity Two A MEASURE PRESERVING TRANSFORMATION WHOSE SPECTRUM HAS LEBESGUE COMPONENT OF MULTIPLICITY TWO J. MATHEW AND M. G. NADKARNI Introduction In this paper we exhibit an ergodic measure preserving transformation on a finite measure space which has Lebesgue component in its spectrum with multiplicity two. Besides this, there is also a discrete part in the spectrum. The question whether there exists an ergodic measure preserving transformation which has Lebesgue component in its spectrum with finite non zero multiplicity was raised by Helson and Parry in their paper [1]. See also Parry [2, p. 50]. Helson and Parry [1] mention also the problem, attributed to Banach, whether there exists an ergodic measure preserving transformation on a finite measure space whose spectrum is simple Lebesgue. In [3, p. 219], Rokhlin mentions the problem of finding an ergodic measure preserving transformation on a finite measure space whose spectrum is Lebesgue type with finite multiplicity. Thus our example, which is given in Section 2, answers the question of Helson and Parry and is a contribution towards the questions of Banach and Rokhlin. We express here our indebtedness to the paper of Helson and Parry. Section 1 DEFINITION. A measure preserving transformation x on a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Measure Preserving Transformation whose Spectrum has Lebesgue Component of Multiplicity Two

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/16.4.402
Publisher site
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Abstract

A MEASURE PRESERVING TRANSFORMATION WHOSE SPECTRUM HAS LEBESGUE COMPONENT OF MULTIPLICITY TWO J. MATHEW AND M. G. NADKARNI Introduction In this paper we exhibit an ergodic measure preserving transformation on a finite measure space which has Lebesgue component in its spectrum with multiplicity two. Besides this, there is also a discrete part in the spectrum. The question whether there exists an ergodic measure preserving transformation which has Lebesgue component in its spectrum with finite non zero multiplicity was raised by Helson and Parry in their paper [1]. See also Parry [2, p. 50]. Helson and Parry [1] mention also the problem, attributed to Banach, whether there exists an ergodic measure preserving transformation on a finite measure space whose spectrum is simple Lebesgue. In [3, p. 219], Rokhlin mentions the problem of finding an ergodic measure preserving transformation on a finite measure space whose spectrum is Lebesgue type with finite multiplicity. Thus our example, which is given in Section 2, answers the question of Helson and Parry and is a contribution towards the questions of Banach and Rokhlin. We express here our indebtedness to the paper of Helson and Parry. Section 1 DEFINITION. A measure preserving transformation x on a

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1984

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