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In [[mathematics]], in the study of [[fractal]]s, a '''Hutchinson operator''' (also known as the '''Barnsley Operator'''<ref>Parallel Processing and Applied Mathematics: 7th International Conference ... By Roman Wyrzykowski</ref>) is a collection of functions on an underlying space ''E''. The [[Iterated function|iteration]] of these functions gives rise to the attractor of an [[iterated function system]], for which the fixed set is [[self-similar]].
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==Definition==
Formally, let <math>\{f_i : X \to X | 1\leq i \leq N\}</math> be an [[iterated function system]], or a set of ''N'' [[contraction mapping|contractions]] from a [[compact set]] ''X'' to itself. We may regard this as defining an operator ''H'' on the power set '''P''' ''X'' as
 
:<math>H : A \mapsto \bigcup_{i=1}^N f_i[A],\,</math>
 
where ''A'' is any subset of ''X''.
 
A key question in the theory is to describe the fixed sets of the operator ''H''. One way of constructing such a fixed set is to start with an initial point or set ''S''<sub>0</sub> and iterate the actions of the ''f''<sub>''i''</sub>, taking ''S''<sub>''n''+1</sub> to be the union of the images of ''S''<sub>n</sub> under the operator ''H''; then taking ''S'' to be the union of the ''S''<sub>''n''</sub>, that is,
 
:<math>S_{n+1} = \bigcup_{i=1}^N f_i[S_n] </math>
 
and
 
:<math>S = \lim_{n \to \infty} S_n . </math>
 
==Properties==
Hutchinson (1981) considered the case when the ''f''<sub>''i''</sub> are [[contraction mapping]]s on a Euclidean space ''X'' = '''R'''<sup>d</sup>.  He showed that such a system of functions has a unique [[Compact space|compact]] (closed and bounded) fixed set ''S''. The proof<ref>{{cite book|last=Sagan|first=Hans|title=Space filling curves|year=1994|publisher=Springer|location=New York ;Berlin [u.a.]|isbn=0-387-94265-3}}</ref> consists in showing that the Hutchinson operator itself is a contraction mapping on the set of compact subsets of ''X'' (endowed with the [[Hausdorff distance]]).
 
The collection of functions <math>f_i</math> together with composition form a [[monoid]]. With ''N'' functions, then one may visualize the monoid as a full [[k-ary tree|N-ary tree]] or a [[Cayley tree]].
 
==See also==
*[[John E. Hutchinson]]
*[[Michael Barnsley]]
 
==Notes==
<references />
 
==References==
* {{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 }}
* {{cite book | title=Chaos and Fractals: New Frontiers of Science | author=Heinz-Otto Peitgen | authorlink=Heinz-Otto Peitgen | coauthors=Hartmut Jürgens, Dietmar Saupe | year=2004 | isbn=0-387-20229-3 | pages=84,225 | publisher=[[Springer-Verlag]] }}
 
{{DEFAULTSORT:Hutchinson Operator}}
[[Category:Fractals]]

Latest revision as of 02:11, 7 May 2014

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