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Math. Z. 223, 643-649 (1996) Mathematische

Summary: Math. Z. 223, 643-649 (1996)
ZeitschriftSpringer-Verlag 1996
One-dimensional chain recurrent sets of flows
in the 2-sphere
K. Athanassopoulos
Departmentof Mathematics,Universityof Crete, GR-71409Iraklion,Greece
e-mail: athanako@talos.cc.uch.gr
Received 19 September 1994;in finalform27 February 1995
1. Introduction
The subject of the classical Poincare-Bendixson theory is the study of the struc-
ture of the limit sets of flows in the 2-sphere S 2 and the behavior of the orbits
near them. A fairly complete account of the theory is given in [3]. A limit set
of a flow in S 2 which contains at least one nonsingular point is 1-dimensional,
compact, connected, invariant and the restricted flow on it is chain recurrent. The
motivation of this note was to examine what properties of limit sets can be ex-
tended to the class of 1-dimensional invariant chain recurrent continua for flows
in S 2. It seems that some basic properties do extend. For instance, an assertion
similar to the Poincare-Bendixson theorem is true in this wider class. Precisely,
if a 1-dimensional invariant chain recurrent continuum of a flow in S 2 contains


Source: Athanassopoulos, Konstantin - Department of Mathematics, University of Crete


Collections: Mathematics