Isolating neighborhood


In the theory of dynamical systems, an isolating neighborhood is a compact set in the phase space of an invertible dynamical system with the property that any orbit contained entirely in the set belongs to its interior. This is a basic notion in the Conley index theory. Its variant for non-invertible systems is used in formulating a precise mathematical definition of an attractor.

Definition

Conley index theory

Let X be the phase space of an invertible discrete or continuous dynamical system with evolution operator
A compact subset N is called an isolating neighborhood if
where Int N is the interior of N. The set Inv consists of all points whose trajectory remains in N for all positive and negative times. A set S is an isolated invariant set if S = Inv for some isolating neighborhood N.

Milnor's definition of attractor

Let
be a discrete dynamical system. A compact invariant set A is called isolated, with isolating neighborhood N if A is the intersection of forward images of N and moreover, A is contained in the interior of N:
It is not assumed that the set N is either invariant or open.