Homeomorphisms between Limbs of the Mandelbrot Set

Details Homeomorphisms between Limbs of the Mandelbrot Set Homeomorphisms between Limbs of the Mandelbrot Set

1995-05-14

arxiv.org/abs/math/9505207v1

Mathematics

Dynamical Systems

Scientific paper

Given $p/q$ and $p'/q$ both irreducible, we construct homeomorphisms between the $p/q$ and the $p'/q$ limbs of the Mandelbrot set. This homeomorphisms are not compatible with the dynamics. Moreover, the filled Julia sets of corresponding parameter values are also homeomorphic. All the homeomorphisms above have counterparts on the combinatorial level relating corresponding external arguments, in the dynamical planes as well as in the parameter spaces. Assuming local connectivity of $M$ we may conclude that the constructed homeomorphisms between limbs are compatible with the embeddings of the limbs in the plane.

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