Isolating neighborhood

Definition

Conley index theory

Let X be the phase space of an invertible discrete or continuous dynamical system with evolution operator

: F_t: X\to X, \quad t\in\mathbb{Z}, \mathbb{R}.

A compact subset N is called an isolating neighborhood if

: \operatorname{Inv}(N,F):={x\in N: F_t(x)\in N{\ }\text{for all }t} \subseteq \operatorname{Int}, N,

where Int N is the interior of N. The set Inv(N,F) consists of all points whose trajectory remains in N for all positive and negative times. A set S is an isolated (or locally maximal) invariant set if S = Inv(N, F) for some isolating neighborhood N.

Milnor's definition of attractor

Let

: f: X\to X

be a (non-invertible) discrete dynamical system. A compact invariant set A is called isolated, with (forward) isolating neighborhood N if A is the intersection of forward images of N and moreover, A is contained in the interior of N:

: A=\bigcap_{n\geq 0}f^{n}(N), \quad A\subseteq\operatorname{Int}, N.

It is not assumed that the set N is either invariant or open.

References

• Konstantin Mischaikow, Marian Mrozek, Conley index. Chapter 9 in Handbook of Dynamical Systems, vol 2, pp 393–460, Elsevier 2002