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On minimal sets in 2-manifolds By Konstantin Athanassopoulos and Polychronis Strantzalos at Athens 1. The structure of limit sets is important to understand the phase portraits of flows. A serious step in this direction is the study of minimal sets, at least because every compact invariant set contains such a set. It is known that a compact minimal set of a C 2 -differentiable dynamical System on a 2-manifold M is either a fixed point, a periodic orbit or eise all of M, in which case the system is an irrational flow on the torus, see A.J. Schwartz [17] (note that the compactness of the manifold is not necessary, see P. Hartman [12], Ch. VII, 12. 1). Such a minimal set will be called simple in the sequel. The assumption in Schwartz's theorem that the dynamical system is at least C 2differentiable is essential, äs it had been shown by A. Denjoy [8]. This paper is concerned with the question of how the qualitative behavior near a compact minimal set of a continuous but not necessarily differentiable flow affects the structure of this minimal set. We intend to prove the following Theorem. A compact minimal set of a dynamical system on a 2-manifold is simple, if it is either stable or a

Published

1988

Language

EN