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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 ﬁlledin 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.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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