In mathematics, in the study of fractals, a Hutchinson operator (also known as the Barnsley Operator) 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.
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,
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 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).
- Parallel Processing and Applied Mathematics: 7th International Conference ... By Roman Wyrzykowski
- Sagan, Hans (1994). Space filling curves. New York ;Berlin [u.a.]: Springer. ISBN 0-387-94265-3.