FractionalGeometry-Chap2

tn a t a b moreover for any compact set d r2 lim

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Unformatted text preview: lying again the kind of calculation introduced in Fig. 2.15, we see h(A1 , B1 ) = 1/4, h(A2 , B2 ) = 1/8, and h(A1 ∪ A2 , B1 ∪ B2 ) = 1/4. Consequently, h(A1 ∪ A2 , B1 ∪ B2 ) = max{h(A1 , B1 ), h(A2 , B2 )}. Is the result of Example 2.3.2 always true? Remember, proofs are near the end of the section, just before the Practice problems. Lemma 2.3.1 h(U ∪ V, W ∪ X ) ≤ max{h(U, W ), h(V, X )}. Now we relate the h-contraction factor of T viewed as function T : K(R2 ) → K(R2 ) to the d-contraction factor of T viewed as a function T : R2 → R2 . Lemma 2.3.2 If T : R2 → R2 is a d-contraction with contraction factor r, then for all compact sets A and B in R2 , h(T (A), T (B )) ≤ r · h(A, B ). 46 CHAPTER 2. ITERATED FUNCTION SYSTEMS Combining Lemmas 2.3.1 and 2.3.2, we show T is an h-contraction, and deduce its contraction factor. First, we shall see what Example 2.3.1 shows about this issue. Recall h(A, B ) = 1 and h(T (A), T (B )) = 1/4. It is easy to see the d-contraction factors of T...
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