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Hutchinson operator

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

Contents

Definition

Let { f i : X X   |   1 i 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 X as

H ( S ) = 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 X (which can be a single point, called a seed) and iterate H as follows

S n + 1 = H ( S n ) = i = 1 N 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

Hutchinson operator Wikipedia


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