Unformatted text preview: ) ∈ T 2 (x0 , y0 ).
Prob 2.4.3 From the proof of Theorem 2.4.1, show for any compact set S ⊂ R2
and for any contraction T : R2 → R2 , diam(T (S )) ≤ r · diam(S ).
Prob 2.4.4 Complete the proof of A ⊆ OI + (w0 , z0 ) in Cor. 2.4.2 in the case
j < k.
Problems 5 through 8 deal with the four transformations of Example 2.4.1.
Prob 2.4.5 (a) Find the coordinates of the point with address 111 · · · = (1)∞ .
Hint: Show this point is the ﬁxed point of T1 .
(b) Find the coordinates of the point with address 2(1)∞ . Find the coordinates
of the point with address 1(2)∞ .
(c) Find the coordinates of the point with address (12)∞ . Find the coordinates
of the point with address (21)∞ .
Prob 2.4.6 In the unit square S , sketch all those subsquares of address length
3 that have empty interiors if the compositions T1 ◦ T2 , and T2 ◦ T1 are forbidden.
Prob 2.4.7 In the unit square S , write Sn3 for the subsquare with address a
sequence of n 3s.
(a) Compute the Hausdorﬀ distances h(S3 , S33 ) and h(S33 , S333 ).
(b) Compute h(Sn3 , S(n+1)3 ) for all n > 1.
(c) What is limn→∞ Sn3 ? (It is a familiar geometric object.)
Prob 2.4.8 In the unit square...
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