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
- [math]\displaystyle{ F_t: X\to X, \quad t\in\mathbb{Z}, \mathbb{R}. }[/math]
A compact subset N is called an isolating neighborhood if
- [math]\displaystyle{ \operatorname{Inv}(N,F):=\{x\in N: F_t(x)\in N{\ }\text{for all }t\} \subseteq \operatorname{Int}\, N, }[/math]
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
- [math]\displaystyle{ f: X\to X }[/math]
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:
- [math]\displaystyle{ A=\bigcap_{n\geq 0}f^{n}(N), \quad A\subseteq\operatorname{Int}\, N. }[/math]
It is not assumed that the set N is either invariant or open.
See also
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
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