Hutchinson operator explained

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

\{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 asH(S) = c up ^{N} _{i =1} 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 followsS _{n+1} = H(S _{n)} = c up ^{N} _{i =1} f _{i(S} _{n)}

and taking the limit, the iteration converges to the attractor

A = \lim _{n} 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.

References

1. Hutchinson . John E. . Fractals and self similarity . Indiana Univ. Math. J. . 30 . 1981 . 713–747 . 10.1512/iumj.1981.30.30055 . 5 . free .
2. Barnsley . Michael F. . Stephen Demko . Iterated function systems and the global construction of fractals . Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences . 399 . 1985 . 243–275 . 1817 .