FractionalGeometry-Chap2

# Prxssol 293 from our experience in the rst two

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Unformatted text preview: four iterates of the right IFS of Fig. 2.57. This list of types is not such a surprise: within these categories only one more possibility remains: disconnected and containing non-contractible loops. There appear to be no attractors of this type, at least using three functions of the form (2.18). The hybrid type was not mentioned on pg 237 of [125]. The be sure we haven’t imagined a nonexistent type of shape, observe in this hybrid IFS that T1 (x, y ) = (x/2, −y/2) + (0, 1/2) T3 (x, y ) = (x/2, y/2) + (0, 1/2) Denoting by L the line segment between (0, 0) and (0, 1), we see L = T1 (L) ∪ T3 (L) Because it is invariant under T1 and T3 , L is contained in the attractor A of {T1 , T2 , T3 }, and so A is not totally disconnected. On the other hand, Fig. 2.58 shows the ﬁrst four iterations of this IFS applied to the illed-in unit square. This shows A is not connected. Recalling (2.19), we can always take s = 1/2. Consequently, the orientation of the image of each of the three transformations is determin...
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