Homeomorphisms of the Mandelbrot Set

Abstract

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 non-trivial homeomorphisms h : M → M. It is shown that these groups have the cardinality of ℝ, and they are not compact.