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

: $F_t: X\to X, \quad t\in\mathbb, \mathbb.$

A compact subset ''N'' is called an isolating neighborhood if

: $\operatorname(N,F):=\ \subseteq \operatorname\, N,$

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'':

: $A=\bigcap_f^(N), \quad A\subseteq\operatorname\, N.$

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 i
''Handbook of Dynamical Systems''
vol 2, pp 393–460, Elsevier 2002
* {{Scholarpedia, title=Attractor, urlname=Attractor, curator=John Milnor

Limit sets