FractionalGeometry-Chap2

To avoid a thicket of special cases we make the

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Unformatted text preview: on of the limit set. The limit set of any subset of Euclidean space is closed. Corollary 2.4.2 The limit set of the orbit OI + (w0 , z0 ) generated by any implementation of the random IFS applied to (w0 , z0 ) ∈ A is the attractor A of the / IFS. Fig. 2.26 ilustrates Cor. 2.4.2. Here 5000 points are generated by two renderngs of the random IFS algorithm, both starting from a point outside the gasket. The first few forward orbit points are circled. Of course, none of the points of the forward orbit lies on the gasket, but these points approach the gasket arbitrarily closely, each about half as far from the gasket as its predecessor. Early implementations of IFS software didn’t find a fixed point for an initial point, but used any point, often selected by mouse click. At the resolution of the monitor, a clean picture of the attractor could be obtained by not plotting the first 20 or 30 points of the forward orbit. What happens if the transformations are not applied randomly? This...
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