FractionalGeometry-Chap2

In addition although well develop the random ifs

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Unformatted text preview: , and T3 (x, y ) = (x/2, y/2) + (0, 1/2), and T = T1 ∪ T2 ∪ T3 . Finally, denote by S the unit square S = {(x, y ) : 0 ≤ x, y ≤ 1}. (a) Compute the Hausdorff distance h(S, G ). (b) Compute the Hausdorff distance h(T n (S ), G ) for all n > 2. (c) Compute limn→∞ h(T n (S ), G ). Prob 2.3.4 Continuing with Prob. 2.3.3, given any ǫ > 0, show how to find a large enough N for which n > N and m > N implies h(T n (S ), T m (S )) < ǫ. That is, show the {T n (S )} is a Cauchy sequence in the Hausdorff metric. Prob 2.3.5 Define T : K(R2 ) → K(R2 ) by T (A) = {(2x, 2y ) : (x, y ) ∈ A}. Is T continuous in the Hausdorff metric? That is, for every A ∈ K(R2 ) and every ǫ > 0, find a δ > 0 for which h(A, B ) < δ implies h(T (A), T (B )) < ǫ. If T is not continuous, find a specific C ∈ K(R2 ) at which continuity fails. 2.4 Convergence of random IFS The deterministic IFS algorithm works just fine to render an image of the attractor of an IFS de...
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