New spectral multiplicities for ergodic actions. (English) Zbl 1254.37003

Solomko, Anton V.

Stud. Math. 208, No. 3, 229-247 (2012).

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 \(T\).

Reviewer: Alexander Kachurovskij (Novosibirsk)

MSC:

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

Keywords:

spectral multiplicity; ergodic action; (C,F)-construction; Poisson suspension