FractionalGeometry-Chap2

# 15 left to right a and b b a1 a b12 and b a12 example

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Unformatted text preview: ween A and B is the smallest ǫ for which both A ⊆ Bǫ and B ⊆ Aǫ hold. Looking more closely at the thickenings, we see for all ǫ < 1, B ⊆ Aǫ , and so h(A, B ) = 1. See Fig. 2.15. To ﬁnd the smallest ǫ for which A ⊆ Bǫ , locate a point a′ ∈ A that maximizes {d(a′ , b) : b ∈ B }, and for this a′ , locate a point b′ ∈ B that minimizes d(a′ , b′ ). (Compactness of A and B guarantee these maxima and minima are realized by points of A and B .) This minimum d(a′ , b′ ) is the smallest ǫ. 36 CHAPTER 2. ITERATED FUNCTION SYSTEMS A B Figure 2.15: Left to right. A and B , B ⊆ A1 , A ⊆ B1/2 , and B ⊆ A1/2 . Example 2.2.1 Convergence in the Hausdorﬀ distance: the product of two Cantor sets Figure 2.16: The product of two Cantor sets. The product A of two Cantor sets shown in Fig. 2.16 is generated by this IFS r 1/4 1/4 1/4 1/4 s 1/4 1/4 1/4 1/4 θ 0 0 0 0 ϕ 0 0 0 0 e 0 3/4 0 3/4 f 0 0 3/4 3/4 Starting with the ﬁlled-in unit square A0 , let A1 = T1 (A0 ) ∪ T2 (A...
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## This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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