On v ‐Distal Flows on 3‐Manifolds

In [ 2 ], H. Furstenberg studied a distal action of a locally compact group G on a compact metric space X , and established a structure theorem. As a consequence, he showed that if G is abelian, then a simply connected space X does not admit a minimal distal G ‐action. In this paper we concern ourselves with a nonsingular flow φ = {φ t } on a closed 3‐manifold M . Recall that φ is called distal if for any distinct two points x , y ∈ M , the distance d (φ t x , φ t y ) is bounded away from 0. The distality depends strongly upon the time parametrization. For example, there exists a time parametrization of a linear irrational flow on T 2 which yields a nondistal flow [ 4, 6 ]. 1991 Mathematics Subject Classification 58F25, 57R30.