Summary: Comment. Math. Helv. 72 (1997) 618635
0010-2571/97/040618-18 $ 1.50+0.20/0
c 1997 Birkh¨auser Verlag, Basel
Commentarii Mathematici Helvetici
A class of flows on 2-manifolds with simple recurrence
Konstantin Athanassopoulos, Theodoros Petrescou and Polychronis Strantzalos
Abstract. We study D-stable flows on orientable 2-manifolds 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 2-manifold of finite genus supporting a non-minimal D-stable
flow must be homeomorphic to an open subset of the 2-sphere or the 2-torus. In the case of the
presence of singularities we necessarily have an open subset of the 2-sphere.
Mathematics Subject Classification (1991). 58F25, 54H20.
Keywords. D-stable flow, 2-manifold of finite genus, recurrence, periodic orbit, local center.
The main object of study in this article is the class of D-stable flows on 2-manifolds
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 D-stable flows is proved to be suitable for the
main purposes of the two theories.