FractionalGeometry-Chap2

# On the other hand for 1 a b 1 1 exercises prob

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Unformatted text preview: he Hausdorﬀ metric h. Given an IFS {T1 , . . . , Tn }, Prop. 2.3.2 shows the collage map T : K(R2 ) → 2.3. CONVERGENCE OF DETERMINISTIC IFS 49 K(R2 ) is a contraction in the Hausdorﬀ metric h. Consequently, the Contraction Mapping Theorem guarantees there is a unique ﬁxed point A for T . This is the attractor A of Theorem 2.3.1. Practice problems Prxs 2.3.1 For the IFS {T1 , T2 } with T1 , T2 : R → R deﬁned by T1 (x) = x/2 and T2 (x) = x/4 + 3/4, with I = [0, 1], and collage map T : K(R) → K(R), (a) Compute h([0, 1], T ([0, 1])). (b) Compute h(T ([0, 1]), T 2 ([0, 1])). (c) For all n, compute h(T n ([0, 1]), T n+1 ([0, 1])). (d) For all n and m, compute h(T n ([0, 1]), T n+m ([0, 1])). Prxs 2.3.2 For all sets A and B in R and for all δ &gt; 0, must (A ∩ B )δ = Aδ ∩ Bδ . Practice problem solutions PrxsSol 2.3.1 (a) First, T ([0, 1]) ⊆ [0, 1] = ([0, 1])0 . Next, the gap in T ([0, 1]) has length 1/4, and this gap must be ﬁlled in order for an ǫ-nbhd (T ([0, 1]))ǫ to cover [0,...
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