Hutchinson operator

In mathematics, in the study of fractals, a Hutchinson operator is a collection of functions on an underlying space "E". The iteration on these functions gives rise to an iterated function system, for which the fixed set is self-similar.

Definition

Formally, let "f"_"i" be a finite set of "N" functions from a set "X" to itself. We may regard this as defining an operator "H" on the power set P "X" as

:

where "A" is any subset of "X".

A key question in the theory is to describe the fixed sets of the operator "H". One way of constructing such a fixed set is to start with an initial point or set "S"₀ and iterate the actions of the "f"_"i", taking "S"_"n"+1 to be the union of the images of "S"_n under the operator "H"; then taking "S" to be the union of the "S"_"n", that is,

:

and

:

Properties

Hutchinson (1981) considered the case when the "f"_"i" are contraction mappings on a Euclidean space "X" = R^d. He showed that such a system of functions has a unique compact (closed and bounded) fixed set "S".

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

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