Rational parameter rays of the Mandelbrot set

TL;DR

This paper presents a new combinatorial proof that all rational external rays land on the Mandelbrot set, linking external angles to dynamics, and introduces partitions and symbolic representations of the set.

Contribution

It offers a novel combinatorial approach to prove ray landing, avoiding complex analytic dependence, and extends understanding of hyperbolic components and symbolic dynamics.

Findings

All rational external rays land on the Mandelbrot set.

Introduces partitions of dynamical and parameter planes.

Interprets the Mandelbrot set via kneading sequences and internal addresses.

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.