FractionalGeometry-Chap2

# 32 shows the collage map t kr2 23 convergence of

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 } &lt; 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),...
View Full Document

## This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

Ask a homework question - tutors are online