Hutchinson operator


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 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 and iterate as follows
and taking the limit, the iteration converges to the attractor

Properties

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.