It is shown that for each N > 0 and for a wide class of Abelian non-compact locally compact second countable groups G including all infinite countable discrete ones and R d 1 × Z d 2 with d 1 , d 2 ≥ 0, there exists a weakly mixing probability preserving G-action with a homogeneous spectrum of multiplicity N .
In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.
We prove that the absolute value of the slope is a (measure theoretic) invariant in the class of von Neumann special flows with one discontinuity, i.e. two ergodic von Neumann flows with one discontinuity are not isomorphic if the slopes of the roof functions have different absolute values, regardless of the irrational rotation in the base.
We prove that special flows over an ergodic rotation of the circle under a C 1 roof function with one discontinuity do not have local rank one. In particular, any such flow has infinite rank.
Let G be a locally compact second countable Abelian group. Given a measure preserving action T of G on a standard probability space (X, µ), let 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, µ) ⊖ C by U T (g)f := f •T −g . In the case when G is either a discrete countable Abelian group or R n , n 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 N are realizable as M(T ) for a weakly mixing G-action T .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.