Wolf Jung
Families of Homeomorphic Subsets of the Mandelbrot Set
(476K, gzipped PostScript)

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/ .