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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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