New spectral multiplicities for ergodic actions

Volume 208 / 2012

Anton V. Solomko Studia Mathematica 208 (2012), 229-247 MSC: 37A15, 37A30. DOI: 10.4064/sm208-3-3


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 $T$.


  • Anton V. SolomkoInstitute for Low Temperature Physics & Engineering
    National Academy of Sciences of Ukraine
    47 Lenin Ave.
    Kharkov, 61164, Ukraine

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