Algorithmic version for Szemerédi regularity partition

2012-11-28T22:22:04Z

131.193.14.92: /* Formal statement of the regularity lemma */

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 (topology)|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> F_t: X\to X, \quad t\in\mathbb{Z}, \mathbb{R}. </math>

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

: <math> \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>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> 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 ==

* [[Limit set]]

== References ==

* Konstantin Mischaikow, Marian Mrozek, ''Conley index''. Chapter 9 in [http://www.sciencedirect.com/science/handbooks/1874575X ''Handbook of Dynamical Systems''], vol 2, pp 393–460, Elsevier 2002 ISBN 978-0-444-50168-4
* {{Scholarpedia|title=Attractor|urlname=Attractor|curator=[[John Milnor]]}}

[[Category:Limit sets]]

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Algorithmic version for Szemerédi regularity partition