FractionalGeometry-Chap2

# Prxs 243 consider the function t r2 r2 given by t x y

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Unformatted text preview: hat is, (xj −1 , yj −1 ) ∈ Aij−1 ...i1 1 and (xj , yj ) ∈ Aij ij−1 ...i1 1 . Randomness guarantees all character strings of all lengths eventually will occur in the sequence producing Σ. In particular, all strings of length k will occur, and because each new term is inserted at the left-most address position, points of Σ will visit all Aik ...i1 , including of course Ai′ ...i′ , containing (x, y ). 1 k Proof of Corollary 2.4.2. In the proof of Theorem 2.4.1, we take (x0 , y0 ) ∈ A, and generate a sequence of points {(x0 , y0 ), (x1 , y1 ), (x2 , y2 ), . . . } by (xk , yk ) = Tik (xk−1 , yk−1 ) for a random sequence I = {i1 , i2 , i3 , . . . }. Now we generate another sequence of points {(w0 , z0 ), (w1 , z1 ), (w2 , z2 ), . . . } = OI + (w0 , z0 ), with the same sequence I . Then d((xk , yk ), (wk , zk )) = d(Tik . . . Ti1 (x0 , y0 ), Tik . . . Ti1 (w0 , z0 )) ≤ rik · · · ri1 d((x0 , y0 ), (w0 , z0 )) ≤ (max{ri })k d((x0 , y0 ), (w0 , z0 )), (2.12) 60 CHAPTER 2. ITERATED FUNCTION SYSTEMS First we show...
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## This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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