In mathematics, in the study of fractals, a Hutchinson operator is the collective action of a set of contractions, called an iterated function system. The iteration of the operator converges to a unique attractor, which is the often self-similar fixed set of the operator.

Definition

Let be an iterated function system, or a set of contractions from a compact set X to itself. The operator H is defined over subsets S\subset X as A key question is to describe the attractors A=H(A) of this operator, which are compact sets. One way of generating such a set is to start with an initial compact set (which can be a single point, called a seed) and iterate H as follows and taking the limit, the iteration converges to the attractor

Properties

Hutchinson showed in 1981 the existence and uniqueness of the attractor A. The proof follows by showing that the Hutchinson operator is contractive on the set of compact subsets of X in the Hausdorff distance. 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.