FractionalGeometry-Chap2

For an ifs t1 tn we call a nite ordered according

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Unformatted text preview: e order of the points produced in this way? 213213... 321321... 132132... Figure 2.28: Left: 1000 points of the gasket random IFS, with 50 points (in bold) generated by applying T1 , T2 , and T3 in that order. Middle: the next 50 points. Right: the points determined by addresses (132)∞ , (213)∞ , and (321)∞ That we don’t seem to see 50 bold points on the left suggests a sense of convergence. The middle of the figure shows (as small circles) an additional 50 points starting from the last generated bold point on the left. It appears that applying T1 , T2 , then T3 and repeating produces a sequence converging to three points. How are we to understand this? Carefully consider the left side of Fig. 2.28. Repeating the approach illustrated in Fig. 2.26, we have not taken a starting point in the attractor A, but it is inside the unit square S . So we shall keep track of the locations of the points within the images of S . We see (x0 , y0 ) ∈ S , (x1 , y1 ) ∈ S1 , (x2 , y2 ) ∈ S21 , (x3 , y3 ) ∈ S321 , (x4 , y4 ) ∈ S13...
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