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.
Let X be the phase space of an invertible discrete or continuous dynamical system with evolution operator
Ft:X\toX, t\inZ,R.
A compact subset N is called an isolating neighborhood if
\operatorname{Inv}(N,F):=\{x\inN:Ft(x)\inN{ }forallt\}\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.
Let
f:X\toX
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=capn\geqfn(N), A\subseteq\operatorname{Int}N.
It is not assumed that the set N is either invariant or open.