The Global Structure of Iterated Function Systems Page: 3
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For the purposes of this paper, let (X, d) be a compact metric space, and let A
be a finite alphabet. We will also primarily consider X to be a real Euclidean space.
Infinite iterated function systems have been studied by Mauldin and Urbanski in [8]
and [9].
1.1. Topological Background
One of the most important concepts in the theory of iterated function systems is
the concept of a contraction. A contraction is a function f : X - X such that there
exists a real number 0 < r < 1 so that for any x, y e X, d(f(x), f(y)) < r. d(x, y).
Such an r will be called a contraction factor of the function f. Note that f is a special
case of a Lipschitz function, therefore it follows that f is uniformly continuous. Also
note that if f is a contraction, then there is a least r which satisfies the contraction
factor condition. For the rest of this paper, when we refer to the contraction factor
of a contraction, we are refering to this least r.
The following theorem is essential to to theory of iterated function systems, and
so a proof is given of this result. Note, however, that this is a widely known result
and can be found in any introductory level topology text book.
THEOREM 1.1 (Banach Fixed Point Theorem). Iff f: X - X is a contraction map
with contraction factor r, then there exists a unique point xo E X with f(xo) = Xo.
Furthermore, for any x E X and any n E N, we have d(f(n)(x), xo) < rn. d(x, xo)
where
f"(n)(x) = fofno ...of_ (x).
n times
PRooF. For x e X, consider the sequence {x~}~> where and x = f(1)(xn_1) for
n > 1. It is clear that {xo}> is a Cauchy sequence.
Since (X, d) is a complete metric space and the sequence {xs}~> is Cauchy, we
may let x0 E X so that xz - x0 as n 0 oc. We now show that x0 is the unique fixed
point of the contraction f : X - X.
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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/m1/9/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .