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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.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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