FractionalGeometry-Chap2

Here are examples of nonrandom binary innite

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Unformatted text preview: y2 )) : (x1 , y1 ), (x2 , y2 ) ∈ S }. The notion of diameter makes sense for any subset of R2 , if we replace max with sup. For compact sets, the sup is realized by the distance between a pair of points, and so in this case max suffices for the more general sup. For example, 2.4. CONVERGENCE OF RANDOM IFS 53 both A = {(x, y ) : x2 + y 2 < 1} and B = {(x, y ) : x2 + y 2 ≤ 1} have diameter √ 2, while C = {(x, y ) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} has diameter 2. So long as each Ti is a contraction – certainly true in this example – we see |Ti1 Ti2 · · · Tin (A)| → 0 as n → ∞. Then we have Corollary 2.4.1 Each point x ∈ A has an infinite address i1 i2 . . . determined by x ∈ Ti1 Ti2 · · · (B ). To emphasize this, we write x = xi1 i2 . . . . Note the order of the address digits is determined by the composition of the transformations: Aij = Ti (Tj (A)). Another way to view this is as a sort of relative coordinates; that is, Aij is the j part of Ai . This process of subdivision can be continued indefinitely, for each k ≥ 1 yielding 4k subsets Ai1 i2 ...ik of address length k . These sets a...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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