Hutchinson operator

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


 * $$H : A \mapsto \bigcup_{i=1}^N f_i[A],\,$$

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,


 * $$S_{n+1} = \bigcup_{i=1}^N f_i[S_n] $$

and


 * $$S = \bigcup_{n=0}^\infty S_n . $$

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 $$f_i$$ 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.