FractionalGeometry-Chap2

# That is mindx0 y0 x y x y a 0 0 recall

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Unformatted text preview: 21 , (x5 , y5 ) ∈ S21321 , (x6 , y6 ) ∈ S321321 , and so on. Now these points need not lie in the attractor (the gasket), but from Cor. 2.4.2 we know the Sik ...i1 get closer and closer to the attractor as k increases. In fact, √ the k → ∞ limit of Sik ...i1 is a point of A. (Recall diam(Sik ...i1 ) = 2 · 2−k .) From the pattern of subscripts of the Sik ...i1 shown above, we see the sequence {(xi , yi )} approaches three points, having addresses (321)∞ , (132)∞ , and (213)∞ . These points are shown on the right side of Fig. 2.28. Finally, can we determine the coordinates of these points without resorting to numerical approxmiation? Let us say (ξ1 , η1 ) has address (132)∞ , (ξ2 , η2 ) has address (213)∞ , and (ξ3 , η3 ) has address (321)∞ . Observe T2 (ξ1 , η1 ) has 58 CHAPTER 2. ITERATED FUNCTION SYSTEMS address 2(132)∞ = (213)∞ . That is, T2 (ξ1 , η1 ) = (ξ2 , η2 ). Similarly, T3 (ξ2 , η2 ) = (ξ3 , η3 ), and T1 (ξ3 , η3 ) = (ξ1 , η1 ). From these relations, we see the (ξi , ηi ) are ﬁx...
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## This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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