The Global Structure of Iterated Function Systems Page: 17
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For the nth iterate, ()(T), we would remove the middle upside down triangle
from each of the remaining triangles from the (n - 1)8st iterate. This process will
approach the set, in the Hausdorff distance, to the set shown in fig. 1.4.
FIGURE 1.4. The attractor of the IFS {T; pl, c2, 3}
1.3. Applications of Iterated Function Systems
In this section we give a brief discussion of one of the possible applications of
iterated functions systems. This application is to the area of image compression.
The following theorem, the Collage Theorem, which was originally proven by Michael
Barnsley, is an important theorem to theory of iterated function systems, and so its
proof is given here. The statement of the following theorem and corollary along with
there proofs can be found in [3].
THEOREM 1.18 (Collage Theorem). Let {X; 1, cp2, ..., n} be any iterated function
system with contraction factor r and attractor J. For any non-empty set E E /C(X)
dH(E, J) < dH E, (E) -
dOysglinai= 1 ' 1
PROOF. By using the triangle inequality for the Hausdorff metric and the definition
of the attractor of an iterated function system, we have
dH(E,6f) < dH E, i()+dH i()
i= 1 i=1
i= 1 i= 1 i= 117
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Snyder, Jason Edward. The Global Structure of Iterated Function Systems, dissertation, May 2009; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc9917/m1/23/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .