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

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

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