Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account
| dc.creator | Milnor, John W. | |
| dc.date | 1999-05-26 | |
| dc.date.accessioned | 2026-07-07T05:29:15Z | |
| dc.date.available | 2026-07-07T05:29:15Z | |
| dc.description | 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. | |
| dc.description | 51 pages, 25 PostScript figures | |
| dc.identifier | https://arxiv.org/abs/math/9905169 | |
| dc.identifier | http://arxiv.org/abs/math/9905169 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78566 | |
| dc.subject | Dynamical Systems | |
| dc.title | Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account | |
| dc.type | text |