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 hcontraction factor of T viewed as function T : K(R2 ) →
K(R2 ) to the dcontraction factor of T viewed as a function T : R2 → R2 .
Lemma 2.3.2 If T : R2 → R2 is a dcontraction 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 hcontraction, 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 dcontraction factors of T...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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