FractionalGeometry-Chap2

Randomness guarantees all character strings of all

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Unformatted text preview: we give attention to one detail. For an IFS {T1 , . . . , Tn }, we call a finite (ordered according to composition of transformations) sequence {ik , . . . , i1 }, each ij ∈ {1, . . . , n}, principal if it is not made up of repetitions of a smaller sequence. For instance, {1, 2, 3, 1, 2, 1} is principal, while {1, 2, 3, 1, 2, 3} is not. Proposition 2.4.2 If the transformations of an IFS are applied in order of repeating a principal sequence {in , . . . , i1 }, then the points generated converge to exactly n points, lying in the attractor. These points have addresses (in . . . i1 )∞ , (i1 in . . . i2 )∞ , . . . , and (in−1 . . . i1 in )∞ Proofs Proof of Lemma 2.4.1. Suppose (x0 , y0 ) is the fixed point of T1 . Now observe T (x0 , y0 ) = T1 (x0 , y0 ) ∪ T2 (x0 , y0 ) ∪ ... ∪ Tn (x0 , y0 ) = (x0 , y0 ) ∪ T2 (x0 , y0 ) ∪ ... ∪ Tn (x0 , y0 ) so (x0 , y0 ) ∈ T (x0 , y0 ). Iterating this argument it is easy to see (x0 , y0 ) ∈ T k (x0 , y0 ) for all k ≥ 1. (See Exercise 2.4.2.) Now if (x0 , y0 ) ∈ A, the compactness...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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