 
Summary: C O L L O Q U I U M M A T H E M A T I C U M
VOL. 104 2006 NO. 2
REMARKS ON THE REGION OF ATTRACTION OF AN
ISOLATED INVARIANT SET
BY
KONSTANTIN ATHANASSOPOULOS (Iraklion)
Abstract. We study the complexity of the flow in the region of attraction of an
isolated invariant set. More precisely, we define the instablity depth, which is an ordinal
and measures how far an isolated invariant set is from being asymptotically stable within
its region of attraction. We provide upper and lower bounds of the instability depth in
certain cases.
1. Introduction. A fundamental problem in the theory of dynamical
systems is to study the topological and dynamical structure of a compact
invariant set of a continuous flow, and in particular a compact minimal set,
and describe the behavior of the orbits near it. The simplest behavior occurs
near an asymptotically stable compact invariant set A of a continuous flow
on a separable, locally compact, metrizable space M. In this case, if W is
the region of attraction of A, then the flow in W \ A is parallelizable. If
moreover M is a finitedimensional manifold, then A has the shape of a
compact polyhedron (see [7] and [8]).
