mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, in the study of

fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...

s, a Hutchinson operator is the collective action of a set of contractions, called an

iterated function system In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981. IFS fractals ...

. The

iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...

of the operator converges to a unique

attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain c ...

, which is the often

self-similar In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar ...

fixed set of the operator.

Definition

Let

\

be an

, or a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of contractions from a

compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...

X

to itself. The operator

H

is defined over subsets

S\subset X

as :

H(S) = \bigcup_^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_ = H(S_n) = \bigcup_^N f_i(S_n)

and taking the limit, the iteration converges to the attractor :

A = \lim_ 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 In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty set, non-empty compact space, compact subsets o ...

. The collection of functions

f_i

together with composition form a

monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...

. With ''N'' functions, then one may visualize the monoid as a full N-ary tree or a

Cayley tree In statistical mechanics and mathematics, the Bethe lattice (also called a regular tree) is an infinite symmetric regular tree where all vertices have the same number of neighbors. The Bethe lattice was introduced into the physics literature by ...

References

{{DEFAULTSORT:Hutchinson Operator Fractals