FractionalGeometry-Chap2

# Iterated function systems 331 341 431 31 441 41 321

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Unformatted text preview: j ≥ k , we can take k = j and apply the triangle inequality: d((x′ , y ′ ), (wj , zj )) ≤ d((x′ , y ′ ), (xj , yj )) + d((xj , yj ), (wj , zj )) &lt; δ/2 + δ/2 = δ That is, (wj , zj ) ∈ C and consequently 0 &lt; d((x, y ), (wj , zj )) &lt; diam(B ) &lt; ǫ Because ǫ &gt; 0 was arbitrary, this shows (x, y ) is a limit point of OI + (w0 , z0 ). (The case j &lt; k is Exercise 2.4.4.)) To see Λ(OI + (w0 , z0 )) ⊆ A, note by Theorem 2.4.1 that for every inﬁnite sequence I ′ and for every (x0 , y0 ) ∈ A, that OI ′ + (x0 , y0 ) ⊂ A. For any limit point (x′′ , y ′′ ) of OI + (w0 , z0 ), there is a subsequence (wpi , zpi ) → (x′′ , y ′′ ). By eq (2.12), for this same subsequence (xpi , ypi ) → (x′′ , y ′′ ). That is, every limit point of OI + (w0 , z0 ) is a limit point of a sequence of points in A. Because A is closed, it contains all its limit points. That is, Λ(OI + (w0 , z0 )) ⊆ A. 2.4. CONVERGENCE OF RANDOM IFS 61 Proof of Proposition 2.4.1. Because i1 → · · · → in is forbidden, we know int(Ain ...i1 ) = ∅. Now fo...
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## This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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