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 f_i 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 f_i, taking S_n+1 to be the union of the images of S_n under the operator H; then taking S to be the union of the S_n, 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 f_i are contraction mappings on a Euclidean space X = R^d. 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.

References

• Hutchinson, John E. (1981). "Fractals and self similarity". Indiana Univ. Math. J. 30: 713–747. DOI:10.1512/iumj.1981.30.30055. Research Blogging.

• Heinz-Otto Peitgen; Hartmut Jürgens, Dietmar Saupe (2004). Chaos and Fractals: New Frontiers of Science, 84,225. ISBN 0387202293.