Rigidity Of A Class Of Smooth Singular Flows On T-2

We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field X on T-2 (sic) {a}, where X is not defined at a is an element of T-2 and X has one critical point which is a center. It follows that the phase space can be decomposed into a (topological disc) D-X and an ergodic component E-X = T-2 (sic) D-X. Let omega(X) be the 1-form associated to X. We show that if vertical bar integral(EX1) d omega(X1) vertical bar not equal vertical bar integral(EX2) d omega(X2 )vertical bar, then the corresponding flows (V-t(X1)) and (v(t)(X2)) are disjoint. It also follows that for every X there is a uniquely associated frequency alpha = alpha(X) is an element of T. We show that for a full measure set of alpha is an element of T the class of smooth time changes of (v(t)(X alpha)) is joining rigid, i.e., every two smooth time changes are either cohomologous or disjoint. This gives a natural class of flows for which the answer to [15, Problem 3] is positive.