Graphics Reference
In-Depth Information
It is easy to check that a condensation transformation on
X
is a contraction
mapping on H(
X
) that has contractivity factor 0 and that its unique fixed point is its
condensation set.
Definition.
If ((
X
,d),W) is an IFS and w is a condensation transformation of
X
, then
((
X
,d),W » {w}) is called an
IFS with condensation
.
22.4.5 Theorem.
(The Collage Theorem) Let (
X
,d) be a complete metric space and
assume that we are given an
L
ΠH(
X
) and an e>0. Let ((
X
,d),W) be any IFS, or IFS
with condensation, so that
Ê
Á
ˆ
˜
£
U
()
d
L
,
w
L
e
,
H
wW
Œ
where d
H
is the Hausdorff metric on H(
X
). If c is the contractivity factor of the IFS
and
A
is its attractor, then
e
(
)
£
d
H
LA
,
,
1
-
c
or equivalently,
1
Ê
Á
ˆ
˜
U
(
)
£
()
()
d
LA
,
c
d
L
,
w
L
for all
L
Œ
H
X
.
H
H
1
-
wW
Œ
Proof.
See [Barn88].
The importance of the Collage Theorem is that it tells us how we can find an IFS
whose attractor is close to a given set. Specifically, it means that if we want to gen-
erate a particular type of shape, say a tree, we do not have to try lots of contraction
mappings at random. On the other hand, one needs to realize that IFSs are not used
to reproduce any specific object. Furthermore, they define an image and not individ-
ual objects in an image. A useful fact is that small changes to the contraction maps
will lead to small changes in the attractor and that the attractor depends in a “con-
tinuous” way on the contraction maps. Changing one contractive map may make parts
of objects appear or disappear in the image.
We return now to the topic of dimension. What is the dimension of the attractors
of IFSs? To answer this question, we first define the “address” of a point in its attrac-
tor. Given an IFS ((
X
,d),W), one of the things one does is take points
x
of
X
and look
at what happens to them under successive transformations via elements of W, that is,
one deals with points
=
(
)( )
y
ww
ooo
...
w
x
,
12
m
where w
i
ΠW. If W consists of the maps w
1
,w
2
,...,w
k
, then
w
i
=
w
i
for some index j
i
and so
y
is completely specified by the sequence of indices j
1
, j
2
,...,
j
m
. It is useful to introduce some notation.
