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

${\displaystyle F_{t}:X\to X,\quad t\in \mathbb {Z} ,\mathbb {R} .}$

A compact subset N is called an isolating neighborhood if

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

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

${\displaystyle f:X\to X}$

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:

${\displaystyle A=\bigcap _{n\geq 0}f^{n}(N),\quad A\subseteq \operatorname {Int} \,N.}$

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

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 ISBN 978-0-444-50168-4
• John Milnor (ed.). "Attractor". Scholarpedia.