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A survey on spectral multiplicities of ergodic actions. (English) Zbl 1261.37009
Author’s abstract: Given a transformation \(T\) of a standard measure space \((X,\mu )\), let \(\mathcal M(T)\) denote the set of spectral multiplicities of the Koopman operator \(U_T\) defined in \(L^2(X,\mu )\ominus \mathbb C\) by \(U_Tf:=f\circ T\). In this survey paper, we discuss which subsets of \(\mathbb N\cup \{\infty \}\) are realizable as \(\mathcal M(T)\) 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.

MSC:
37A30 Ergodic theorems, spectral theory, Markov operators
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
47A35 Ergodic theory of linear operators
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