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.
Definition Let { f i : X → X | 1 ≤ i ≤ N } {\displaystyle \{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 {\displaystyle X} to itself. The operator H {\displaystyle H} is defined over subsets S ⊂ X {\displaystyle S\subset X} as
H ( S ) = ⋃ i = 1 N f i ( S ) . {\displaystyle H(S)=\bigcup _{i=1}^{N}f_{i}(S).\,}
A key question is to describe the attractors A = H ( A ) {\displaystyle 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 {\displaystyle S_{0}\subset X} (which can be a single point, called a seed) and iterate H {\displaystyle H} as follows
S n + 1 = H ( S n ) = ⋃ i = 1 N f i ( S n ) {\displaystyle 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 → ∞ S n . {\displaystyle A=\lim _{n\to \infty }S_{n}.}
Properties Hutchinson showed in 1981 the existence and uniqueness of the attractor A {\displaystyle A} . The proof follows by showing that the Hutchinson operator is contractive on the set of compact subsets of X {\displaystyle X} in the Hausdorff distance. The collection of functions f i {\displaystyle 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.
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