FractionalGeometry-Chap2

Also the nested sequence n n 1 2 3 of unbounded

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Unformatted text preview: The translations of the IFS guarantee that (B3 )1/4 ∩ A1 = ∅, for example. In Fig. 2.22 in light grey we see the ǫ = 1/4nbhds of A1 , A2 , and A3 . Note that B ⊆ (Ai )1/4 for i = 1, 2, 3. In darker grey we see the ǫ = 1/8-nbhds of A1 , A2 , and A3 . Note that B ⊆ (Ai )1/8 for i = 2, 3, but not for i = 1. Similar observations hold for A and the Bi . Consequently, the larger Euclidean contraction factor, 1/4, of the Ti determines the Hausdorff contraction factor of the collage map. That is, h(T (A), T (B )) = 1/4. Before beginning our proof of convergence of IFS, first we recall some facts from analysis. A sequence {A1 , A2 , . . . } of sets is nested if A1 ⊇ A2 ⊇ A3 ⊇ · · · . Standard results about continuous functions and compact sets include 2.3. CONVERGENCE OF DETERMINISTIC IFS 45 A B Figure 2.22: Relating Euclidean and Hausdorff contraction factors. Proposition 2.3.1 (1) A continuous function takes compact sets to compact sets. (2) Contractions are continuous. (3) The union of a finite collection of compact sets is compact. (4) The intersection of a nested sequen...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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