FractionalGeometry-Chap2

# That is ht a t b 14 before beginning our proof

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Unformatted text preview: the 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 h-contraction 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 ǫ-nbhd 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.

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