FractionalGeometry-Chap2

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Unformatted text preview: and deterministic IFS algorithms. Given an inﬁnite sequence I = (i1 i2 · · · ) with each ij ∈ {1, 2, . . . , n}, the forward orbit of (x0 , y0 ) determined by I is OI + (x0 , y0 ) = {(x0 , y0 ), Ti1 (x0 , y0 ), Ti2 Ti1 (x0 , y0 ), . . . } is an implementation of the random IFS algorithm. Theorem 2.4.1 and Cor. 2.4.2 show that all forward orbits have the same closure, and that is the attractor of the IFS. Theorem 2.4.1 The closure of the orbit generated by the any implementation of the random IFS is the attractor of the corresponding deterministic IFS; that is, A is the closure of Σ. As an illustration of Theorem 2.4.1, Fig. 2.25 shows three images of the gasket produced by the random algorithm. In the left and middle pictures the three transformations are applied equally frequently. That is, the probability of applying each Ti is 1/3. In the right side of Fig. 2.25 the probabilities have been modiﬁed: T1 and T2 have been applied with probability 1/5, and T3 with 2.4. CONVERGENCE OF RANDOM IFS 55 probability...
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