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 selfsimilar fixed set of the operator.
Let
{
f
i
:
X
→
X

1
≤
i
≤
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
⊂
X
as
H
(
S
)
=
⋃
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
⊂
X
(which can be a single point, called a seed) and iterate
H
as follows
S
n
+
1
=
H
(
S
n
)
=
⋃
i
=
1
N
f
i
(
S
n
)
and taking the limit, the iteration converges to the attractor
A
=
lim
n
→
∞
S
n
.
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 Nary tree or a Cayley tree.