FractionalGeometry-Chap2

An undergraduate classmate of mine decided to

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Unformatted text preview: re squares of √ side length 2−k , hence of diameter 2 · 2−k . For the study of the random IFS algorithm, the central fact is that the diameters decrease with increasing address length. Suppose we specify a resolution ǫ > 0 and want to render A to within this resolution. That is, we want to approximate A by a finite set of points V = {v1 , v2 , ..., vn } having the property that for every point w ∈ A, there is √ some vj with d(w, vj ) < ǫ. Simply take k large enough that 2 · 2−k < ǫ. Then construct V so it contains at least one point in each Ai1 i2 ...ik . For any point w ∈ A, there is some Ai1 i2 ...ik with w ∈ Ai1 i2 ...ik . Then some vj ∈ Ai1 i2 ...ik , ad √ so d(w, vj ) ≤ diam(Ai1 i2 ...ik ) = 2 · 2−k < ǫ. Now of course these sets are not disjoint, points on some boundaries have multiple addresses. This is another manifestation of the phenomenon whose more familiar example is the nonuniqueness of decimal expansions: 1/10 can be written as 0.1000... and as 0.09999.... (The geometric series 9/(102 ) + 9/(103 ) + · · ·...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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