In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set. The IFS described is composed of contractions whose images, as a collage or union when mapping the given set, are arbitrarily close to the given set. It is typically used in fractal compression.
Let
X
be a complete metric space. Suppose
L
is a nonempty, compact subset of
X
and let
ϵ
≥
0
be given. Choose an iterated function system (IFS)
{
X
;
w
1
,
w
2
,
…
,
w
N
}
with contractivity factor
0
≤
s
<
1
, (The contractivity factor of the IFS is the maximum of the contractivity factors of the maps
w
i
.) Suppose
h
(
L
,
⋃
n
=
1
N
w
n
(
L
)
)
≤
ε
,
where
h
(
⋅
,
⋅
)
is the Hausdorff metric. Then
h
(
L
,
A
)
≤
ε
1
−
s
where A is the attractor of the IFS. Equivalently,
h
(
L
,
A
)
≤
(
1
−
s
)
−
1
h
(
L
,
∪
n
=
1
N
w
n
(
L
)
)
, for all nonempty, compact subsets L of
X
.
Informally, If
L
is close to being stabilized by the IFS, then
L
is also close to being the attractor of the IFS.