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