In mathematics, in the study of fractals, a Hutchinson operator is a collection of functions on an underlying space E. The iteration on these functions gives rise to an iterated function system, for which the fixed set is self-similar.
Definition
Formally, let fi be a finite set of N functions from a set X to itself. We may regard this as defining an operator H on the power set P X as
![H : A \mapsto \bigcup_{i=1}^N f_i[A],\,](http://wpcontent.answers.com/math/4/a/8/4a8a098b33d50165a1581eacd3f05686.png)
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 S0 and iterate the actions of the fi, taking Sn+1 to be the union of the images of Sn under the operator H; then taking S to be the union of the Sn, that is,
![S_{n+1} = \bigcup_{i=1}^N f_i[S_n]](http://wpcontent.answers.com/math/d/d/9/dd9bb7190dc33fd3c408296850b992df.png)
and
Properties
Hutchinson (1981) considered the case when the fi are contraction mappings on a Euclidean space X = Rd. He showed that such a system of functions has a unique compact (closed and bounded) fixed set S.
The collection of functions fi together with composition form a monoid. With N functions, then one may visualize the monoid as a full N-ary tree or a Cayley tree.
References
- Hutchinson, John E. (1981). "Fractals and self similarity". Indiana Univ. Math. J. 30: 713–747. doi:.
- Heinz-Otto Peitgen; Hartmut Jürgens, Dietmar Saupe (2004). Chaos and Fractals: New Frontiers of Science. pp. 84,225. ISBN 0387202293.
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