Homeomorphisms of the Mandelbrot Set
نویسنده
چکیده
On subsets of the Mandelbrot set, EM ⊂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 : EM → EM in the parameter plane. Each h has two fixed points in EM , 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 EM ⊂ M constructed by surgery are extended to homeomorphisms of M, and employed to study groups of nontrivial homeomorphisms h :M→M. It is shown that these groups have the cardinality of R, and they are not compact. Preprint of a paper submitted to Dynamics in the Complex Plane, proceedings of a symposium in honour of Bodil Branner, June 19–21 2003, Holbæk.
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