an:06020741
Zbl 1254.37003
Solomko, Anton V.
New spectral multiplicities for ergodic actions
EN
Stud. Math. 208, No. 3, 229-247 (2012).
00297305
2012
j
37A15 37A30
spectral multiplicity; ergodic action; (C,F)-construction; Poisson suspension
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 \(T\).
Alexander Kachurovskij (Novosibirsk)