## Institute for Mathematical Sciences

## Preprint ims99-3

** J. Milnor**
* 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 angles
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 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.

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