Adrien Douady

A proof of Thurston's topological characterization of rational functions

The criterion proved in this paper was stated by Thurston in November 1982. Thurston lectured on its proof on several occasions, notably at the NSF summer conference in Duluth, 1983, where one of the authors (JHH) was present. Using the notes of various attendants at these lectures, we have reconstructed a proof that we have made as precise as we could. Since this required a certain amount of work on our part, we thought it might be of some use to present this proof to the reader. We thank Dennis Sullivan for useful conversations, and Fritz yon Haesseler and especially Ben Wittner for help with the writing and valuable suggestions. After the first version was written, Clifford Earle pointed out that better estimates than what we had were to be found in [B].

Existence of excited states for a nonlinear Dirac field

AbstractWe prove the existence of infinitely many stationary states for the following nonlinear Dirac equation

Julia Sets and the Mandelbrot Set

i\gamma ^\mu \partial _\mu \psi - m\psi + (\bar \psi \psi )\psi = 0.

On the dynamics of polynomial-like mappings

Seeking for eigenfunctions splitted in spherical coordinates leads us to analyze a nonautonomous dynamical system inR2. The number of eigenfunctions is given by the number of intersections of the stable manifold of the origin with the curve of admissible datum. This proves the existence of infinitely many stationary states, ordered by the number of nodes of each component.

Étude dynamique des polynômes complexes

Publisher Summary This chapter discusses algorithms for computing angles in the Mandelbrot set. The classes having three elements or more is made of rational points, each class with 2 elements is limit of classes made of rational points. The numbers θ_(n) converge to a number θ_(∞) known as—the Morse number. The convergence of θ_(n) to θ_(∞) is faster than any exponential convergence: the number of good digits is doubled at each time; this is because of the growing of hairs. The chapter describes spiraling angle.

Conformally natural extension of homeomorphisms of the circle

Quadratic Julia sets, and the Mandelbrot set, arise in a mathematical situation which is extremely simple, namely from sequences of complex numbers defined inductively by the relation

Implosion parabolique et dimension de Hausdorff

z_n + = z_n^2 + c,