 
Summary: Comment. Math. Helv. 72 (1997) 618635
00102571/97/04061818 $ 1.50+0.20/0
c 1997 Birkh¨auser Verlag, Basel
Commentarii Mathematici Helvetici
A class of flows on 2manifolds with simple recurrence
Konstantin Athanassopoulos, Theodoros Petrescou and Polychronis Strantzalos
Abstract. We study Dstable flows on orientable 2manifolds of finite genus in connection with
the topology of the underlying phase spaces. The description of the phase portrait is used to
prove that a connected orientable 2manifold of finite genus supporting a nonminimal Dstable
flow must be homeomorphic to an open subset of the 2sphere or the 2torus. In the case of the
presence of singularities we necessarily have an open subset of the 2sphere.
Mathematics Subject Classification (1991). 58F25, 54H20.
Keywords. Dstable flow, 2manifold of finite genus, recurrence, periodic orbit, local center.
1. Introduction
The main object of study in this article is the class of Dstable flows on 2manifolds
of finite genus (for definition see section 2). We are concerned with their qualitative
behavior in connection with the topological structure of the underlying manifold.
This point of view is in the center of the theory of transformation groups and
dynamical systems. The class of Dstable flows is proved to be suitable for the
main purposes of the two theories.
