[FOM] Mandelbrot set, Catalan numbers, Context-free grammars

[Rempe-Gillen, Lasse]

Just a remark that there is more to the importance of the Mandelbrot set within one-dimensional dynamics.

Indeed, McMullen proved that the Mandelbrot set is universal in most nontrivial (in a certain precise sense) one-dimensional families of rational maps.

More precisely, he showed that any such family, there are copies of the connectedness locus of the family of "unicritical" polynomials,  z\mapsto z^d+c (which, up to a change of variable,  are exactly the polynomials with exactly one critical value). The Mandelbrot set corresponds to d=2. As multiple critical points (corresponding to the higher-degree "Multibrot" sets) are a non-generic phenomenon, usually what you will see is the Mandelbrot set.

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Professor Lasse Rempe-Gillen
Dept. of Math. Sciences, Univ. of Liverpool, Liverpool L69 7ZL
Office 520; tel. +44 (0)151 794 4025, fax +44 (0)151 794 4061
http://pcwww.liv.ac.uk/~lrempe
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