FractionalGeometry-Chap2

Proof of prop 232 apply the denition of t and observe

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Unformatted text preview: Figure 2.23: Convergence of the iterates of an IFS. Example 2.3.3 Convergence of the deterministic IFS algorithm. Consider the IFS r 1/2 1/2 1/2 s 1/2 1/2 1/2 θ 0 π/2 −π/2 ϕ 0 π/2 −π/2 e 0 1 0 f 0 0 1 47 2.3. CONVERGENCE OF DETERMINISTIC IFS The sequence of sets in Fig. 2.23 illustrates the speed of convergence of prefractals to their fractal limit. Here the initial set B is the filled-in unit square. Note the tiny visible differences between the last two pictures of the figure, T 8 (B ) and T 9 (B ). By now, it should be clear that the Hausdorff distance between successive iterates decreases rapidly. Proofs Proof of Lemma 2.3.1. Let δ1 = h(U, W ) and δ2 = h(V, X ), so U ⊆ Wδ1 , W ⊆ Uδ1 , V ⊆ Xδ2 , and X ⊆ Vδ2 . Writing δ = max{δ1 , δ2 }, we see U ⊆ Wδ , W ⊆ Uδ , V ⊆ Xδ , and X ⊆ Vδ , and consequently U ∪ V ⊆ Wδ ∪ Xδ (2.6) Also, it is not difficult to show (see Exercise 2) that Wδ ∪ Xδ ⊆ (W ∪ X )δ (2.7) From eqs (2.6) and (2.7) we see U ∪ V ⊆ (W ∪ X )δ . Similarly, W ∪ X ⊆ (U ...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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