Xavier Buff

On König's root finding algorithms

In this paper, we first recall the definition of a family of root-finding algorithms known as Konigs algorithms. We establish some local and some global properties of those algorithms. We give a characterization of rational maps which arise as Konigs methods of polynomials with simple roots. We then estimate the number of non-repelling cycles Konigs methods of polynomials may have. We finally study the geometry of the Julia sets of Konigs methods of polynomials and produce pictures of parameter spaces for Konigs methods of cubic polynomials.

Siegel disks with smooth boundaries

We show the existence of angles α ∈ R/Z such that the quadratic polynomial Pα(z) = e2iπαz + z2 has a Siegel disk with C∞-smooth boundary. This result was first announced by R. Perez-Marco in 1993.

Julia Sets in Parameter Spaces

Abstract: Given a complex number λ of modulus 1, we show that the bifurcation locus of the one parameter family {fb(z)=λz+bz2+z3}b∈ℂ contains quasi-conformal copies of the quadratic Julia set J(λz+z2). As a corollary, we show that when the Julia set J(λz+z2) is not locally connected (for example when z↦λz+z2 has a Cremer point at 0), the bifurcation locus is not locally connected. To our knowledge, this is the first example of complex analytic parameter space of dimension 1, with connected but non-locally connected bifurcation locus. We also show that the set of complex numbers λ of modulus 1, for which at least one of the parameter rays has a non-trivial accumulation set, contains a dense Gδ subset of S1.

Quadratic polynomials, multipliers and equidistribution

Given a sequence of complex numbers {\rho}_n, we study the asymptotic distribution of the sets of parameters c {\epsilon} C such that the quadratic maps z^2 +c has a cycle of period n and multiplier {\rho}_n. Assume 1/n.log|{\rho}_n| tends to L. If L {\leq} log 2, they equidistribute on the boundary of the Mandelbrot set. If L > log 2 they equidistribute on the equipotential of the Mandelbrot set of level 2L - 2 log 2.

Teichmüller spaces and holomorphic dynamics

One fundamental theorem in the theory of holomorphic dynamics is Thurstons topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmuller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurstons theorem (the marked Thurstons theorem). We also mention some applications and related results, as well as the notion of deformation spaces of rational maps introduced by A. Epstein.1 The spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 The gluing and the operad structures . . . . . . . . . . . . . . . . . . . . . . 11 3 Framed little discs and the Gerstenhaber and BV Structures . . . . . . . . . 23 4 Moduli space, the Sullivan–PROP and (framed) little discs . . . . . . . . . . 35 5 Stops, Stabilization and the Arc spectrum . . . . . . . . . . . . . . . . . . . 40 6 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7 Open/Closed version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Virtually repelling fixed points

In this article, we study the notion of virtually repelling fixed point. We first give a definition and an interpretation of it. We then prove that most proper holomorphic mappings f : U → V with U contained in V have at least one virtually repelling fixed point.

Geometry of the Feigenbaum map

We show that the Feigenbaum-Cvitanovic equation can be interpreted as a linearizing equation, and the domain of analyticity of the Feigenbaum fixed point of renormalization as a basin of attraction. There is a natural decomposition of this basin which enables to recover a result of local connectivity by Jiang and Hu for the Feigenbaum Julia set.

How regular can the boundary of a quadratic Siegel disk be

In the family of quadratic polynomials with an irrationally indifferent fixed point, we show the existence of Siegel disks with a fine control on the degree of regularity of the linearizing map on their boundary. A general theorem is stated and proved. As a particular case, we show that in the quadratic family, there are Siegel disks whose boundaries are Cn but not C n+1 Jordan curves.

From local to global analytic conjugacies

Let

Algorithmic Construction of Hurwitz Maps

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