The Global Structure of Iterated Function Systems Page: 27
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Now, to cover 6rn, note that diam (Jm) = 22m-1= 22m-r1diam (,(1)) and the left
endpoint of Jm is {22m-1}. Therefore, in order to cover m, we need to stretch pl and
P2 by a factor of 22m-1 and shift them to the right 22m-1 units. But this is exactly
what the contractions ck and pk+1 do for k = 2m + 1. Therefore, for k = 2m + 1, we
have
k(jn) Uk+1() _6mrn
Thus,
2n
i=1
( ()) U 2 (n)) u ... U (21 ( ) U2n (n)
= () UJ6U...jU/-1
Thus, we have shown that j() is indeed the attractor of the iterated function system
{X; y 1, 2** ... 2n4.
Recall that for every p > 0, we have Fp := {1/n }>1 U{0} and for every 0 < r < 1
we defined Cr : {r}>o U {0}.
PROPOSITION 2.6. For every p > 0, Fp is an attractor of an iterated function system.
PROOF. Consider the functions
{ 3P(x)- __1__
0 mll1 (2+ )P
Clearly, P and 3 are contractions. It remains to show that 2 is a contraction and
that 1'(6) 6 /. We will first show that 2 is a contraction.
To that end, note that
2
(2 + P)P+l27
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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/m1/33/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .