FractionalGeometry-Chap2

# So long as each ti is a contraction certainly true in

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Unformatted text preview: n of all of the Ti produce points that stay in A. Lemma 2.4.2 Σ ⊆ A. With Lemma 2.4.2 we have shown that the points generated by the random algorithm all lie in the attractor A of the IFS. Next we show that every sequence Σ produced by the random algorithm comes arbitrarily close to ﬁlling up the attractor A. In computational terms, this means the (ﬁnite) sequence of points generated by an implementation of the random algorithm eventually will ﬁll in the attractor to the resolution of the monitor. Pixels are not points, so eventually every pixel containing points of A will have been visited by the 52 CHAPTER 2. ITERATED FUNCTION SYSTEMS random sequence, and the image on the screen will not change, regardless of the additional runtime. When the Ti are linear, this stabilization of the image can occur rather rapidly; for nonlinear Ti , it can take a very, very long time, and we need to seek other approaches. A central notion, one we shall use in many instances in addition to proving the equivalence of the random and determinis...
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