Attractor blocks

Conley and Easton [4] introduced the concept of an "isolating block" as a tool for the study of the topological properties of isolated invariant sets. A special case is an isolating block for an attractor, also called an "attractor block". A set is an attractor block if its image is strictly interior to itself .

The following two theorems give the correspondence between attractors and attractor blocks. Every attractor block has an attractor in its interior given by allowing the block to converge through successive iterates of the map. The attractor thus obtained is the maximal invariant set inside the block. Conversely, every attractor can be surrounded by an attractor block with the property that the attractor is the maximal invariant set inside the block.

Theorem 3.1 If is an attractor block, then is an attractor.

Proof.

Theorem 3.2 If is an attractor and if is any neighborhood of , then there exists an attractor block such that .

To prove this theorem, we need basically two things: the metric characterization of attractor blocks and the notion of pseudo-orbits. The idea is the following: if we consider the set of all points which can be reached from the attractor by -pseudo-orbits, then for sufficiently small , this set will be an attractor block for the given attractor.

Proof.