FractionalGeometry-Chap2

# FractionalGeometry-Chap2

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Unformatted text preview: 0, all suﬃciently small changes in these parameters produce T ′ (S ) with T ′ (S ) ⊆ (T (S ))ǫ and with T (S ) ⊆ (T ′ (S ))ǫ , that is, h(S, T ′ (S )) ≤ ǫ. 2.9. SOME TOPOLOGY OF IFS ATTRACTORS 93 Figure 2.59: The eﬀects of changing r, s, θ, and ϕ for the shape on the left. C D A B Figure 2.60: Left: the placement of the four transformations. Right: the attractor witout rotations or reﬂections. The eﬀect of the translations is even more straightforward, simply moving the pieces left or right, up or down. Small enough changes in translations leave each piece of the image in an ǫ-nbhd of the unchanged piece. Finally, each fractal is made up of pieces within pieces within pieces, and each parameter change echoes down forever. However, with composition the scalings multiply, and so the cumulative eﬀect is bounded by the sum of a convergent geometric series. The change in parameter distance is the maximum of the parameter changes realized by each transformation, so keeping this small enough bounds the resulting Hausdorﬀ distance changes of the IFS attr...
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## This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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