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New spectral multiplicities for ergodic actions. (English) Zbl 1254.37003
Author’s abstract: Let $$G$$ be a locally compact second countable abelian group. Given a measure preserving action $$T$$ of $$G$$ on a standard probability space $$(X, \mu)$$, let $$\mathcal M(T)$$ denote the set of essential values of the spectral multiplicity function of the Koopman representation $$U_T$$ of $$G$$ defined in $$L^2(X,\mu)\ominus \mathbb C$$ by $$U_T(g)f := f\circ T_{-g}$$. If $$G$$ is either a discrete countable Abelian group or $$\mathbb R^n, n\geq 1$$, it is shown that the sets of the form $$\{p,q,pq\}, \{p,q,r,pq,pr,qr,pqr\}$$ etc. or any multiplicative (and additive) subsemigroup of $$\mathbb N$$ are realizable as $$\mathcal M(T)$$ for a weakly mixing $$G$$-action $$T$$.

##### MSC:
 37A15 General groups of measure-preserving transformations and dynamical systems 37A30 Ergodic theorems, spectral theory, Markov operators
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