Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account

Abstract

A key point in Douady and Hubbard's study of the Mandelbrot set $M$ is the theorem that every parabolic point $c\ne 1/4$ in $M$ is the landing point for exactly two external rays with angle which are periodic under doubling. This note will try to provide a proof of this result and some of its consequences which relies as much as possible on elementary combinatorics, rather than on more difficult analysis. It was inspired by section 2 of the recent thesis of Schleicher (see also Stony Brook IMS preprint 1994/19, with E. Lau), which contains very substantial simplifications of the Douady-Hubbard proofs with a much more compact argument, and is highly recommended. The proofs given here are rather different from those of Schleicher, and are based on a combinatorial study of the angles of external rays for the Julia set which land on periodic orbits. The results in this paper are mostly well known; there is a particularly strong overlap with the work of Douady and Hubbard. The only claim to originality is in emphasis, and the organization of the proofs.

51 pages, 25 PostScript figures

Cited by in corpus (5)

• A Landing Theorem for Periodic Rays of Exponential Maps
• On the inhomogeneity of the Mandelbrot set
• Dynamics of projectable functions: Towards an atlas of wandering domains for a family of Newton maps
• Entire functions with Cantor bouquet Julia sets
• Robust sensitive dependence of geometric Gibbs states for analytic families of quadratic maps