Rigidity times for a weakly mixing dynamical system which are not rigidity times for any irrational rotation

摘要

We construct an increasing sequence of natural numbers (m(n))(n=1)(+infinity) with the property that (m(n)theta[1])(n,1) is dense in T for any theta I RQ, and a continuous measure on the circle mu such that limn ->infinity integral T vertical bar vertical bar m(n)theta vertical bar vertical bar d mu(theta) = 0. Moreover, for every fixed k is an element of N, the set {n is an element of N: k inverted iota m(n)} is infinite. This is a sufficient condition for the existence of a rigid, weakly mixing dynamical system whose rigidity time is not a rigidity time for any system with a discrete part in its spectrum.