 
Summary: Math. Proc. Camb. Phil. Soc. (1990), 108, 569 5 6 9
Printed in Great Britain
The flow near nontrivial minimal sets on 2manifolds
BY KONSTANTIN ATHANASSOPOULOS
Freie Universitdt Berlin, Institut fur Mathematik II {WE 2), Arnimallee 3,
D1000 Berlin 33, Germany
(Received 18 December 1989; revised 6 April 1990)
1. Introduction
In this paper we give a description of the qualitative behaviour of the orbits near
a nontrivial compact minimal set of a continuous flow on a 2manifold. The first
results in this direction were obtained in [1] and the present paper can be regarded
as a continuation of that work. The main result can be stated as follows:
THEOREM 1*1. Let (IR,Jf,/) be a continuous flow on a 2manifold M and A <=M a non
trivial compact minimal set. Then, there exists a connected, open, invariant
neighbourhood E of A with the following properties:
(a) the restricted flow on E\A is completely unstable;
(b) ifxsE, then L+
(x) U L~(x) cA[)dE and L+
(x) = A or L~(x) = A;
(c) every connected component of E\A contains at least one orbit C(x) such that
