External ray

Abstract: Dichotomie dynamique de Fatou et Julia. Points periodiques. Consequences du theoreme de Montel. L'ensemble de Julia est la fermeture de l'ensemble des points periodiques repulseurs. Resultats classiques sur l'ensemble de Fatou. Classification de Sullivan de l'ensemble de Fatou. Une condition pour le developpement sur l'ensemble de Julia. La dynamique des polynomes. L'ensemble de Mandelbrot et le travail de Douady et Hubbard. Le theoreme d'application de Riemann mesurable et la dynamique analytique

Abstract: A key point in Douady and Hubbard's study of the Mandelbrot set $M$ is the theorem that every parabolic point $c e 1/4$ 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.

Abstract: Introduction L.Tan Preface J. Hubbard 1. The Mandelbrot set is universal C. McMullen 2. Baby Mandelbrot sets are born in cauliflowers A. Douady, X. Buff, R. Devaney and P. Sentenac 3. Modulation dans l'ensemble de Mandelbrot P. Haissinsky 4. Local connectivity of Julia sets: expository lectures J. Milnor 5. Holomorphic motions and puzzles (following M. Shishikura) P. Roesch 6. Local properties of the Mandelbrot set at parabolic points L.Tan 7. Convergence of rational rays in parameter spaces C. Petersen and G. Ryd 8. Bounded recurrence of critical points and Jakobson's Theorem S. Luzzatto 9. The Herman-Swiatek theorems with applications C. Petersen 10. Perturbations d'une fonction linearisable H. Jellouli 11. Indice holomorphe et multiplicateur H. Jellouli 12. An alternative proof of Mane's theorem on non-expanding Julia sets M. Shishikura and L.Tan 13. Geometry and dimension of Julia sets Y. -C. Yin 14. On a theorem of Mary Rees for the matings of polynomials M. Shishikura 15. Le theoreme d'integrabilite des structures presque complexes A. Douady and X. Buff 16. Bifurcation of parabolic fixed points M. Shishikura.

Abstract: We describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. Internal addresses are a convenient and efficient way of describing the combinatorial structure of the Mandelbrot set, and of giving geometric meaning to the ubiquitous kneading sequences in human-readable form (Sections 3 and 4). A simple extension, \emph{angled internal addresses}, distinguishes combinatorial classes of the Mandelbrot set and in particular distinguishes hyperbolic components in a concise and dynamically meaningful way. This combinatorial description of the Mandelbrot set makes it possible to derive existence theorems for certain kneading sequences and internal addresses in the Mandelbrot set (Section~6) and to give an explicit description of the associated parameters. These in turn help to establish some algebraic results about permutations of periodic points and to determine Galois groups of certain polynomials (Section~7). Through internal addresses, various areas of mathematics are thus related in this manuscript, including symbolic dynamics and permutations, combinatorics of the Mandelbrot set, and Galois groups.