Hutchinson operator

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In mathematics, in the study of fractals, a Hutchinson operator (also known as the Barnsley Operator[1]) is a collection of functions on an underlying space E. The iteration of these functions gives rise to the attractor of an iterated function system, for which the fixed set is self-similar.

Definition[edit]

Formally, let \{f_i : X \to X | 1\leq i \leq N\} be an iterated function system, or a set of N contractions from a compact set X to itself. We may regard this as defining an operator H on the power set P X as

H : A \mapsto \bigcup_{i=1}^N f_i[A],\,

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 S0 and iterate the actions of the fi, taking Sn+1 to be the union of the images of Sn under the operator H; then taking S to be the union of the Sn, that is,

S_{n+1} = \bigcup_{i=1}^N f_i[S_n]

and

S = \lim_{n \to \infty} S_n .

Properties[edit]

Hutchinson (1981) considered the case when the fi are contraction mappings on a Euclidean space X = Rd. He showed that such a system of functions has a unique compact (closed and bounded) fixed set S. The proof[2] consists in showing that the Hutchinson operator itself is a contraction mapping on the set of compact subsets of X (endowed with 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.

See also[edit]

Notes[edit]

  1. ^ Parallel Processing and Applied Mathematics: 7th International Conference ... By Roman Wyrzykowski
  2. ^ Sagan, Hans (1994). Space filling curves. New York ;Berlin [u.a.]: Springer. ISBN 0-387-94265-3. 

References[edit]