Jung, W.
Homeomorphisms of the Mandelbrot Set
(511K, pdf)

ABSTRACT.  On subsets E of the Mandelbrot set M , homeomorphisms are 
constructed by quasi-conformal surgery. When the dynamics of quadratic 
polynomials is changed piecewise by a combinatorial construction, a general 
theorem yields the corresponding homeomorphism h: E to E in the parameter 
plane. Each h has two fixed points in E , and a countable family of 
mutually homeomorphic fundamental domains. Possible generalizations to 
other families of polynomials or rational mappings are discussed. The 
homeomorphisms on subsets E of M constructed by surgery are extended 
to homeomorphisms of M , and employed to study groups of non-trivial 
homeomorphisms h: M to M . It is shown that these groups have the 
cardinality of the continuum, and they are not compact.