Metric characterization of attractor blocks

This short section is devoted to a metric characterization of attractor blocks.

It will be useful to measure the distance that an attractor block maps inside itself. For , define

Note that is the minimum distance from to and that , for any subset .

Our main result here is the following theorem:

Theorem 3.11 A nonempty compact set is an attractor block if and only if .

Proof.

The following corollary gives a sufficient condition on a subset for its closure to be an attractor block. The proof is an immediate consequence of the previous theorem and Lemma 3.10.

Corollary 3.12 Let be a nonempty subset of . If and if is compact, then is an attractor block.