99-240 Wolf Jung
Families of Homeomorphic Subsets of the Mandelbrot Set (476K, gzipped PostScript) Jun 23, 99
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Abstract. The $1/3$-limb of the Mandelbrot set $M$ is considered as a graph, where the vertices are given by certain Misiurewicz points. The edges are described as a union of building blocks that are called "frames". There is a 1-1 correspondence between these frames and star-shaped subsets of the Julia set $K_a$, where $a$ denotes the Misiurewicz point $\gamma_M(11/56)$. This global combinatorial correspondence between $M_{1/3}$ and $K_a$ provides a complement to the asymptotic similarity obtained by Tan Lei. The frames are defined by recursions for external angles or para-puzzles. By quasi-conformal surgery we construct a homeomorphism $h$ of the edge from $a$ to $\gamma_M(23/112)$ onto itself, and show that the frames on this edge are homeomorphic. This can be generalized to all edges of $\M_{1/3}$, and to all limbs of $\M$. The method is similar to that of Branner-Douady and Branner-Fagella, but the construction of the "first return map" is different. The reader is invited to obtain the related program mandel.exe from http://www.iram.rwth-aachen.de/~jung/ .

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