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A CHARACTERIZATION OF DENJOY FLOWS KONSTANTIN ATHANASSOPOULOS
 

Summary: A CHARACTERIZATION OF DENJOY FLOWS
KONSTANTIN ATHANASSOPOULOS
1
An interesting problem in the theory of dynamical systems is to determine the
global structure of a flow from properties of characteristic invariant subsets of the
phase space such as minimal sets or Poisson stable orbit closures. For flows on
manifolds of dimension greater than two, the behaviour of the flow near such sets is
far from being well understood. In this note we give a characterization of Denjoy
flows on the torus, that is, suspensions of orientation preserving homeomorphisms of
the unit circle onto itself with a Cantor minimal set, via conditions referring to the
asymptotic behaviour of the orbits near a strictly Poisson stable orbit closure. More
precisely, we prove the following.
THEOREM 1.1. A flow (U,M,f) on a closed 2-manifold M is topologically
equivalent to a Denjoy or irrationalflowif and only if there is a strictly Poisson stable
orbit C{x) and an open invariant neighbourhood V of C(x) such that L+
(y) = L~(y)for
every yeV.
This theorem isfirstproved in the particular case where the strictly Poisson stable
orbit is contained in a non-trivial minimal set, using the description of the flow near
non-trivial minimal sets on 2-manifolds given in [2]. The general case is proved using

  

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

 

Collections: Mathematics