A survey on spectral multiplicities of ergodic actions

Abstract

AbstractGiven a transformationTof a standard measure space (X,μ), let ℳ(T) denote the set of spectral multiplicities of the Koopman operatorUTdefined in$L^2(X,\mu )\ominus \Bbb C$byUTf:=f∘T. In this survey paper we discuss which subsets of$\Bbb N\cup \{\infty \}$are realizable as ℳ(T) for variousT: 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.