Spectral multiplicities and asymptotic operator properties of actions with invariant measure

New sets of spectral multiplicities of ergodic automorphisms of a probability space are proposed. Realizations have been obtained, inter alia, for the sets of multiplicities $\{p,q,pq\}$, $\{p,q,r,pq,pr,rq,pqr\}$ and so on. It is also shown that systems with homogeneous spectrum may have factors over which they form a finite extension. Moreover, these systems feature arbitrary polynomial limits, and thus may serve as useful elements in constructions. A so-called minimal calculus of multiplicities is proposed. Some infinite sets of multiplicities occurring in tensor products are calculated, which involve a Gaussian or a Poisson multiplier. Spectral multiplicities are also considered in the class of mixing actions.
Bibliography: 25 titles.