Attractors

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

(1) is a nonempty compact invariant set and
(2) there exists a neighborhood of such that .

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.

(1) is an attractor.
(2) If is any neighborhood of , then there exists a compact neighborhood of such that and .
(3) If is any neighborhood of , then there exists a positively invariant compact neighborhood of such that and .
(4) is asymptotically stable.

Proof.

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.

Proof.

Theorem 2.3 If is an attractor and if is a compact subset of , then .

Proof.