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Homeomorphisms of the Mandelbrot Set

Wolf Jung
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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.

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Wolf Jung. 2003-12-14. Homeomorphisms of the Mandelbrot Set. https://arxiv.org/abs/math/0312279

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