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distance between successive iterates of the deterministic IFS algorithm. Our
remaining task is to establish how and why this sequence of iterates converges.
The ﬁrst step of argument is showing that the collage map is a contraction
in the Hausdorﬀ metric. That is, given an IFS {T1 , ..., Tn } with each Ti a dcontraction having contraction factor ri , we now show T : K(R2 ) → K(R2 ) is
an hcontraction with contraction factor max ri = max{r1 , ..., rn }. To motivate
the max, consider this example.
Example 2.3.1 The collage of Euclidean contractions gives a Hausdorﬀ contraction.
Take A = {(0, y ) : 0 ≤ y ≤ 1} and B = {(x, 0) : 0 ≤ x ≤ 1}, and T the IFS
T1
T2
T3 r
1/4
1/8
1/8 s
1/4
1/8
1/8 θ
0
0
0 ϕ
0
0
0 e
0
7/8
0 f
0
0
7/8 A calculation like that illustrated in Fig. 2.15 shows h(A, B ) = 1. Writing
Ai = Ti (A) and Bi = Ti (B ) (note here A1 = T1 (A), not the ǫ = 1 nbhd of
A; (A1 )ǫ denotes the ǫnbhd of A1 ), it is easy to see h(A1 , B1 ) = 1/4 and
h(A2 , B2 ) = h(A3 , B3 ) = 1/8....
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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