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External ray

About: External ray is a research topic. Over the lifetime, 252 publications have been published within this topic receiving 5243 citations.


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Book
29 Nov 1994
TL;DR: In this article, the authors present a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics, including geometric function theory, quasiconformal mappings, and hyperbolic geometry.
Abstract: Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c. As discovered by Feigenbaum, such a mapping exhibits a repetition of form at infinitely many scales. Drawing on universal estimates in hyperbolic geometry, this work gives an analysis of the limiting forms that can occur and develops a rigidity criterion for the polynomial f. This criterion supports general conjectures about the behavior of rational maps and the structure of the Mandelbrot set.The course of the main argument entails many facets of modern complex dynamics. Included are foundational results in geometric function theory, quasiconformal mappings, and hyperbolic geometry. Most of the tools are discussed in the setting of general polynomials and rational maps.

748 citations

Journal ArticleDOI
TL;DR: In this article, a parametre d'applications de type polynomial is defined and a set of applications of M dans M. Carrottes pour le dessert are presented.
Abstract: Applications de type polynomial. Familles analytiques de telles applications. Resultats negatifs. Familles a un parametre d'applications de degre 2. Petites copies de M dans M. Carrottes pour le dessert

726 citations

Journal ArticleDOI
TL;DR: In this paper, the authors define a dynamique dynamique de Fatou et Julia, which is defined as the dynamique of points periodiques repulseurs of a group of polynomes.
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

535 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the boundary of the Mandelbrot set M has Hausdorff dimension two and that for a generic c E AM, the Julia set of z I > Z 2 + C also has Hhausdorff dimensions two.
Abstract: It is shown that the boundary of the Mandelbrot set M has Hausdorff dimension two and that for a generic c E AM, the Julia set of z I > Z2 + C also has Hausdorff dimension two. The proof is based on the study of the bifurcation of parabolic periodic points.

215 citations

Posted Content
TL;DR: In this paper, the authors provide a proof of Douady and Hubbard's Mandelbrot set theorem, which relies as much as possible on elementary combinatorics, rather than on more difficult analysis.
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.

129 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20212
20203
20191
20181
20176
20166