Solomko A. Measurable rank-one actions and its applications

Purpose of the work is the construction and study of measure preserving abelian group actions with nontrivian spectral and asymptotical properties. The object of research is the spectral multiplisity problem in ergodic theory and the property of 'weak mixing' for infinite measure preserwing flows. Methods used in the work are methods of calculus and functional analysis. The results obtained are new. The main result are the following. It is shown that for a wide class of abelian non-compact locally compact second countable groups including all infinite countable discrete ones, reals, there exists a weakly mixing probability preserving action with a homogeneous spectrum of arbitrary given multiplicity. 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 N are realizable as the set of spectral multiplicities for weakly mixing actions of such groups. A rank-one infinite measure preserving flow with infinite ergodic index is constructed.