Homeomorphisms of the Mandelbrot Set

Details Homeomorphisms of the Mandelbrot Set Homeomorphisms of the Mandelbrot Set

2003-12-14

arxiv.org/abs/math/0312279v1

Dynamics on the Riemann Sphere, A Bodil Branner Festschrift, P. G. Hjorth, C. L. Petersen Eds., EMS January 2006, pp 139-159.

Mathematics

Dynamical Systems

Preprint of a paper submitted to Dynamics in the Complex Plane, proceedings of a symposium in honour of Bodil Branner, June 19

Scientific paper

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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