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.
Let be an iterated function system, or a set of contractions from a compact set to itself. The operator is defined over subsets as
A key question is to describe the attractors 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 as follows
and taking the limit, the iteration converges to the attractor
Hutchinson showed in 1981 the existence and uniqueness of the attractor . The proof follows by showing that the Hutchinson operator is contractive on the set of compact subsets of in the Hausdorff distance.
The collection of functions 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.