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

TL;DR

This paper provides an elementary combinatorial proof of a key result in the study of the Mandelbrot set, showing that each non-1/4 parabolic point is the landing point of exactly two periodic external rays, emphasizing clarity and organization.

Contribution

It offers a new, combinatorial approach to proving the landing of external rays on parabolic points in the Mandelbrot set, differing from prior analytical methods.

Findings

Every non-1/4 parabolic point in the Mandelbrot set is landing point of exactly two periodic external rays.

The proofs rely on combinatorial analysis of external ray angles and their landing properties.

The approach simplifies understanding of complex dynamics in the Mandelbrot set.

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.