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 $$\{f_i : X \to X\ |\ 1\leq i \leq N\}$$ 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


 * $$H(S) = \bigcup_{i=1}^N f_i(S).\,$$

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 $$S_0\subset X$$ (which can be a single point, called a seed) and iterate $$H$$ as follows


 * $$S_{n+1} = H(S_n) = \bigcup_{i=1}^N f_i(S_n) $$

and taking the limit, the iteration converges to the attractor


 * $$A = \lim_{n \to \infty} S_n . $$

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.