FractionalGeometry-Chap2

The next example shows that we can get adequate

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Unformatted text preview: 0 ) ∪ T3 (A0 ) ∪ T4 (A0 ) = T (A0 ), and continue with An+1 = T (An ). Pictured in Fig. 2.17 are A0 , A1 , A2 , and A3 . Figure 2.17: Iterates converging to the product of two Cantor sets. The containments A0 ⊃ A1 ⊃ A2 ⊃ · · · ⊃ A 37 2.2. THE HAUSDORFF METRIC are clear. Moreover, it is not difficult to see A ⊂ (A0 )0 and A0 ⊂ A√2/4 , A ⊂ (A1 )0 √ 2/4 √ so h(A1 , A) = 2/42 and A1 ⊂ A√2/42 , so h(A0 , A) = ... A ⊂ (An )0 and An ⊂ A√2/4n+1 , so h(An , A) = √ 2/4n+1 Consequently, limn→∞ h(A, An ) = 0. This is the sense in which the sequence A0 , A1 , · · · converges to the fractal A. This calculation is easy. The next example shows that we can get adequate estimates in situations with geometry sufficiently complex to make soberingly difficult (and unnecessary) the exact calculation of Hausdorff distance. Example 2.2.2 Convergence in the Hausdorff distance: the Koch curve The Koch curve K is generated by this IFS r 1/3 1/3 1/3 1/3 s 1/3 1/3 1/3 1/3 θ 0 π/3 −π/3 0 ϕ 0 π/3 −π/3 0 e 0 1/3 1/2 2/3 f 0 0 √ 3...
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