FractionalGeometry-Chap2

For all sets a b r prove a b a b or nd a

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Unformatted text preview: 21, 31, and 41 each give rise to four forbidden triples, 121, 221, 321, and 421, for example. But note that 11 gives no new forbidden addresses, because 111, 211, 311, and 411 are subsquares of 11, 21, 31, and 41. Some care must be exercised in counting empty addresses. PrxsSol 2.4.3 (a) Because A is compact, diam(A) is realized by the Euclidean distance d((x1 , y1 ), (x2 , y2 )) for some pair of points (x1 , y1 ), (x2 , y2 ) ∈ A. For every pair of points (x1 , y1 ), (x2 , y2 ) ∈ A d(T (x1 , y1 ), T (x2 , y2 )) = (y1 − y2 )2 (x1 − x2 )2 + 4 9 62 CHAPTER 2. ITERATED FUNCTION SYSTEMS 331 341 431 31 441 41 321 131 141 421 231 11 241 21 121 221 Figure 2.30: Empty length 3 addresses for Prxs 2.4.2. and so (x1 − x2 )2 (y1 − y2 )2 + ≤ 9 9 ≤ (x1 − x2 )2 (y1 − y2 )2 + 4 9 2 (y1 − y2 )2 (x1 − x2 ) + 4 4 Because this is holds for all pairs of points in A, and the diameter of T (A) is the distance between some pair of points T (x1 , y1 ), T (x2 , y2 ), we have 1 1 diam(A) ≤ diam(T (A)) ≤ diam(A) 3 2 (b) Take AU = {(x, 0) : 0 ≤ x ≤ 1} and AL = {(0, y ) : 0 ≤ y ≤ 1}, both...
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