FractionalGeometry-Chap2

# A find the smallest for which g is simply connected

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Unformatted text preview: S table of PrxsSol 2.2.2. f 0 0 0 1/3 1/3 2/3 2/3 2/3 Figure 2.21: The ﬁrst three stages if the solution to PrxsProb. 2.2.2 Exercises Prob 2.2.1 (a) Find the Hausdorﬀ distance between the unit circle and the square with vertices (1, 1), (−1, 1), (−1, −1), and (1, −1). (b) Find the √ Hausdorﬀ √ distance between the unit circle and the√ square with ver√ √ √ √ √ tices ( 2/2, 2/2), (− 2/2, 2/2), (− 2/2, − 2/2), and (− 2/2, − 2/2). Prob 2.2.2 Is there a compact set A for which (Aδ1 )δ2 = Aδ1 +δ2 ? Is (Aδ1 )δ2 = Aδ1 +δ2 true for all compact sets A? Prob 2.2.3 Complete the proof of Prop. 2.2.1 by showing the Hausdorﬀ distance is symmetric. 2.2. THE HAUSDORFF METRIC 43 Prob 2.2.4 Denote by C the Cantor middle thirds set, the subset of R generated by the IFS given by T1 (x) = x/3 and T2 (x) = x/3 + 2/3. (a) Find the smallest ǫ for which Cǫ has no gaps; that is, Cǫ is connected. (b) Find the smallest ǫ for which Cǫ has exactly one gap; that...
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