FractionalGeometry-Chap2

A recalling the relation between determinants and

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Unformatted text preview: r all in+1 , int(Ain+1 in ...i1 ) = int(Tin+1 (Ain ...i1 )) = Tin+1 (int(Ain ...i1 )) = ∅. The result follows by continuing through in+m . Practice problems Prxs 2.4.1 Find the coordinates of the points (122)∞ , (221)∞ , and (212)∞ with T1 (x) = x/3 and T2 (x) = x/3 + 2/3. Prxs 2.4.2 For the IFS (2.10) sketch the address length 3 subsquares left empty if the compositions T1 ◦ T1 , T2 ◦ T1 , T3 ◦ T1 , and T4 ◦ T1 are forbidden. Prxs 2.4.3 Consider the function T : R2 → R2 given by T (x, y ) = (x/2, y/3). (a) For every compact set A ⊂ R2 , find upper and lower bounds on diam(T (A)) in terms of diam(A). (b) Find sets AU and AL for which the upper and lower bounds are realized. Practice problem solutions PrxsSol 2.4.1 Say x1 has address (122)∞ , x2 has address (212)∞ , and x3 has address (221)∞ . Then notice T1 (T2 (T2 (x1 ))) has address 122(122)∞ = (122)∞ , and so x1 is the fixed point of T1 ◦ T2 ◦ T2 . Solving x1 = T1 ◦ T2 ◦ T2 (x1 ) = ((x1 /3 + 2/3)/3 + 2/3)/3 = x1 /27 +...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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