The following definition is a modification to this setting of that given by Conley [2,3] for flows on compact metric spaces.
Definition: A set
is an attractor for
if
Some authors would call an "attracting set" and would reserve the name "attractor" for an attracting set with further properties. However, in this paper Conley's terminology will be followed.
The preceeding definition is somewhat weak in the sense that the only assumption on the neighborhood is that . However, this assumption is actually very strong. For example, the neighborhood can be taken to be compact, positively invariant , and arbitrarily close to , as stated in Theorem 2.1 below. Also, the notion of attractor corresponds to the more classical notion of "asymptotically stable" .
We have the following theorem:
Theorem 2.1 If
is a nonempty compact
invariant set, then the following statements are equivalent.
The next important notion is the domain of attraction .
The following theorems establish that the domain of attraction of , is an open set and that every compact subset of satisfies .
Theorem 2.2 If
is a compact positively
invariant
neighborhood of an attractor
such that
, then
and hence
is open.
Theorem 2.3 If is an attractor and if is a compact subset of , then .
Copyright (c) 1998 by Richard
McGehee, all rights reserved.