FractionalGeometry-Chap2

Then we write t s t1 s t2 s t3 s 27 21 iterated

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Unformatted text preview: ) = (0, 1/2) and S = {(x, y ) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}, the unit square. Then we write T (S ) = T1 (S ) ∪ T2 (S ) ∪ T3 (S ) 27 2.1. ITERATED FUNCTION SYSTEM FORMALISM T3 S S T1 S T2 S Figure 2.2: The collage map T applied to the unit square. See Fig. 2.2. If we start with any compact set C , say the cat picture on the left side of Fig. 2.3, then the sequence T (C ), T 2 (C ), . . . converges to the gasket. The sense in which this convergence occurs is the topic of Sect. 2.2. ··· Figure 2.3: The collage map T applied to a cat, then repeated. We show that every IFS determines a unique nonempty compact set A characterized by T (A) = A, and with the appropriate interpretation of convergence, exhibiting the remarkable property that for any compact set B , T k (B ) → A as k → ∞. That is, under repeated application of T , all features of the initial set B disappear in the limit and only A remains. In this sense, A is equivalent to the IFS rules. Because T k (B ) → A, A is called the attractor of the IFS. Before proving this convergence theo...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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