Hutchinson 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.

References

References

1. Hutchinson, John E.. (1981). "Fractals and self similarity". Indiana Univ. Math. J..
2. Barnsley, Michael F.. (1985). "Iterated function systems and the global construction of fractals". Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences.