Reading: pg. 407-456
1. (Refer to pg. 173 of the text.) Pick two points in the Cantor set which have nonterminating triadic expansions. Verify that the Hutchinson map W(x) maps them back into the Cantor set. What are the pre-images of your points (that is, those points in the Cantor set which W maps into your points)?
2. (Refer to pg. 277 of the text.) Let P=(x,y) and Q=(u,v) be points in the real plane, and define d(P,Q)=max{1.25|x-u|,|y-v|}. Show that d is a metric on the real plane, and sketch the unit circle with respect to this metric.
3.Use the Fixed Point Theorem to prove that the two maps w1(.) and w2(.) which appear in the Hutchinson map for the Cantor set each converge to a unique fixed point under iteration. What are those fixed points? Why are they different from the fixed point of the Hutchinson map W(.) which is formed from the union of these maps? Show that W is a contraction mapping and hence has a unique fixed point. (Carefully identify X, d, H(X), h, and s.) Comment on the difference between a transformation on X and a transformation on H(X).
4.Write the Hutchinson map for the Sierpinski carpet and show that it is a contraction mapping on (H(R2),h(l2)). Is it also a contraction mapping on (R2,l2)?
5. For the metric space H(X), we defined the Hausdorff distance between two sets. If we add the empty set to H(X), how should we adjust Hausdorff distance? (In other words, if A is an element of H(X), how far is it from the empty set?)
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