FractionalGeometry-Chap2

292 f 12 sin2 34 certainly for 0 the attractor

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Unformatted text preview: ed by the values of r and θ = ϕ. There are 8 possibilities (r, θ) = (+, 0), (+, π/2), (+, π ), (+, 3π/2), (−, 0), (−, π/2), (−, π ), (−, 3π/2), so each of these IFS tables is determined by a point in an 8 × 8 × 8 lattice. A map of those lattice points that give each topological type might reveal interesting patterns, but so far this approach has not been pursued. So far as we know, the first general attempt at determining the topological type of gasket relatives from their IFS parameters was made by Taylor [157]. Proofs Sketch of the proof of Thm 2.9.1: Though it is tedious to find a precise relation between small changes in a parameter and changes in the image of the transformation with modified parameter, the basic idea is illustrated in Fig. 2.59. On the left is the image of the unit square S under a transformation T . Continuing to the right, in turn we increase r, s, θ, and ϕ and compare T (S ) with the corresponding T ′ (S ), in bold. Certainly, for any ǫ >...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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