[For the entire collection see Zbl 0627.00017.]
In a now famous paper S. Schwartzman [Ann. Math., II. Ser. 66, 270- 284 (1957; Zbl 0207.226)] introduced the notion of asymptotic cycles. The idea is to characterize the trajectories of a continuous flow on a manifold in terms of their representations in the singular homology of the manifold. Specifically, let M denote a Riemannian manifold and let \(\phi_ t\) be a continuous flow on M. Given a point \(p\in M\) and a real number T define a 1-cycle \(\gamma_{T,p}\) by taking the trajectory \(\phi_ t(p)\) through p to \(\phi_ t(p)\) then return to p along a minimal geodesic. This loop defines a homology class \([\gamma_{T,p}]\) in \(H_ 1(M,R)\). The asymptotic cycle is \(A(p)=\lim_{T\to \infty}(1/T)[\gamma_{T,p}]\in H_ 1(M,R)\), provided the limit exists.
In the present paper the author considers continuous flows defined on closed orientable 2-manifolds such that the flow has only a finite number of equilibria. The limit sets of the trajectories of such a flow are identified and in all cases the associated asymptotic cycle is computed. In the special case of the 2-torus it is shown that the asymptotic cycles of such a flow exist at every point.