FractionalGeometry-Chap2

32 shows the collage map t kr2 23 convergence of

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Unformatted text preview: (B ) = T (B ). Applying T to this inclusion gives T (B ) ⊇ T (T (B )) = T 2 (B ). Continuing to apply T and then combining the inclusions gives B ⊇ T (B ) ⊇ T 2 (B ) ⊇ T 3 (B ) ⊇ · · · and so B, T (B ), T 2 (B ), . . . is a nested sequence of nonempty sets. By parts (1), (2), and (3) of Prop. 2.3.1, this is a nested sequence of nonempty compact sets, and so by part (4) of Prop. 2.3.1, A= ∞ k=1 T k (B ) (2.8) is a nonempty, compact set. To see A = T (A), note that because the T k (B ) are nested, for each N +1 T (∩N=1 T k (B )) = T (T N (B )) = T N +1 (B ) = ∩N=1 T k (B ) k k Taking the limit as N → ∞, and using continuity of T to interchange T and the limit, we obtain T (A) = A. To prove the uniqueness of A, suppose there is another compact set C with T (C ) = C . Then h(A, C ) = h(T (A), T (C )) ≤ r · h(A, C ) by Prop. 2.3.2. Now r = max{r1 , ..., rn } < 1, so h(A, C ) ≤ r · h(A, C ) requires h(A, C ) = 0; that is, A = C. Finally, let D be any compact set in R2 . Then h(T (D), A) = h(T (D),...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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