 
Summary: manuscripta math. 97, 37 44 (1998) c SpringerVerlag 1998
Konstantin Athanassopoulos
Volume preserving flows with cyclic winding numbers
groups and without periodic orbits on 3manifolds
Received: 17 January 1998 / Revised version: 31 March 1998
Abstract. We construct examples of volume preserving nonsingular C1 vector fields on
closed orientable 3manifolds, which have cyclic winding numbers groups with respect to
the preserved volume element, but have no periodic orbits.
1. Introduction
A great amount of work in dynamical systems has been oriented towards find
ing conditions which guarantee the existence of periodic orbits for homeo
morphisms and flows. The best known condition concerning orientation pre
serving homeomorphisms of the unit circle is the rationality of the PoincarŽe
rotation number. A generalization to continuous flows on closed orientable
2manifolds is given in [2], based on the notions of asymptotic cycle and
winding numbers group of a flow with respect to an invariant Borel probabil
ity measure. The rational rotation numbers are replaced in the general case
by the cyclic winding numbers groups, i.e. winding numbers groups which
are isomorphic to Z.
In the same paper it is shown that for any n 3 there is a smooth flow on
