Hutchinson operator

In mathematics, in the study of fractals, a Hutchinson operator^[1] is the collective action of a set of contractions, called an iterated function system.^[2] The iteration of the operator converges to a unique attractor, which is the often self-similar fixed set of the operator.

Definition

Let ${\displaystyle \{f_{i}:X\to X\ |\ 1\leq i\leq N\}}$ be an iterated function system, or a set of contractions from a compact set ${\displaystyle X}$ to itself. The operator ${\displaystyle H}$ is defined over subsets ${\displaystyle S\subset X}$ as

${\displaystyle H(S)=\bigcup _{i=1}^{N}f_{i}(S).\,}$

A key question is to describe the attractors ${\displaystyle A=H(A)}$ of this operator, which are compact sets. One way of generating such a set is to start with an initial compact set ${\displaystyle S_{0}\subset X}$ (which can be a single point, called a seed) and iterate ${\displaystyle H}$ as follows

${\displaystyle S_{n+1}=H(S_{n})=\bigcup _{i=1}^{N}f_{i}(S_{n})}$

and taking the limit, the iteration converges to the attractor

${\displaystyle A=\lim _{n\to \infty }S_{n}.}$

Properties

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

The collection of functions ${\displaystyle f_{i}}$ 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.