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

$F_t:X\to X,t\in \mathbb{Z},ℝ.$

A compact subset N is called an isolating neighborhood if

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

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:

$A=\bigcap _{n\ge 0}{f}^n(N),A\subseteq \mathrm{Int}N.$

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

See also

• Limit set

References

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