Hutchinson operator

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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[edit]

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[edit]

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[edit]

  1. ^ Hutchinson, John E. (1981). "Fractals and self similarity". Indiana Univ. Math. J. 30 (5): 713–747. doi:10.1512/iumj.1981.30.30055. 
  2. ^ 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.