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A survey on spectral multiplicities of ergodic actions

Published online by Cambridge University Press:  09 December 2011

ALEXANDRE I. DANILENKO
ALEXANDRE I. DANILENKO*
Affiliation:
Institute for Low Temperature Physics & Engineering of National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov, 61164, Ukraine (email: [email protected])
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Abstract

Given a transformation T of a standard measure space (X,μ), let ℳ(T) denote the set of spectral multiplicities of the Koopman operator UT defined in by UTf:=f∘T. In this survey paper we discuss which subsets of are realizable as ℳ(T) for various T: ergodic, weakly mixing, mixing, Gaussian, Poisson, ergodic infinite measure-preserving, etc. The corresponding constructions are considered in detail. Generalizations to actions of Abelian locally compact second countable groups are also discussed.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 81 - 117
Copyright
Copyright © Cambridge University Press 2011

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