FractionalGeometry-Chap2

is cauchy if for every 0 there is an n for which i j

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Unformatted text preview: is, Cǫ has two components. (c) Find the smallest ǫ for which Cǫ has exactly three gap; that is, Cǫ has four components. (d) Find the smallest ǫ for which Cǫ has 2n components. Prob 2.2.5 Denote by G the right isosceles Sierpinski gasket, the attractor of the IFS Ti (x, y ) = (x/2, y/2) + (ei , fi ) where (e1 , f1 ) = (0, 0), (e2 , f2 ) = (1/2, 0), and (e3 , f3 ) = (0, 1/2). (a) Find the smallest ǫ for which Gǫ is simply-connected; that is, the thickened gasket has no holes. (b) Find the smallest ǫ for which Gǫ has one hole. (c) Find the smallest ǫ for which Gǫ has 1 + 3 holes. (d) Find the smallest ǫ for which Gǫ has 1 + 3 + 9 holes. (e) Find the smallest ǫ for which Gǫ has 1 + 3 + · · · + 3n holes. Prob 2.2.6 For all r, 0 < r < 1, denote by Cr the Cantor middle rth set: a subset of [0, 1] obtained as the limit of successively removing the middle rth of each interval. That is, Cr is the attractor of the IFS T1 (x) = ((1 − r)/2)x and T2 (x) = ((1 − r)/2)x + (1 + r)/2. (a) For general r, compute h([0, 1], Cr ). (b) Find h(C1/4 , C1/2 ). (c) Find a general...
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