Rational parameter rays of the Mandelbrot set

Abstract

We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is different from the original one, given by Douady and Hubbard and refined by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic dependence of the polynomials with respect to parameters and can thus be made to apply for non-complex analytic parameter spaces; this proof is also technically simpler. Finally, we derive several corollaries about hyperbolic components of the Mandelbrot set. Along the way, we introduce partitions of dynamical and parameter planes which are of independent interest, and we interpret the Mandelbrot set as a symbolic parameter space of kneading sequences and internal addresses.

33 pages, 9 figures

References in corpus (1)

• Dynamics in one complex variable: introductory lectures

Cited by in corpus (14)

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• Core entropy and biaccessibility of quadratic polynomials
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• Combinatorial rigidity for unicritical polynomials
• The Thurston Algorithm for quadratic matings
• Accessible Boundary Points in the Shift Locus of a Familiy of Meromorphic Functions with Two Finite Asymptotic Values
• The topology of deformation spaces of Kleinian groups
• Quadratic matings and ray connections
• Convergence of Rays with Rational Argument in Hyperbolic Components
• Parapuzzle of the Multibrot set and typical dynamics of unimodal maps
• Homeomorphisms of the Mandelbrot Set