Tan Lei and Shishikura's example of non-mateable degree 3 polynomials without a Levy cycle

After giving an introduction to the procedure dubbed slow polynomial mating and stating a conjecture relating this to other notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree 3 post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree 6 rational map. We give a conjectural interpretation of this in terms of pinched spheres and show further conformal representations.Comment: 35 page

Herman's condition and critical points on the boundary of Siegel disks of polynomials with two critical values

We extend a theorem of Herman from the case of unicritical polynomials to the case of polynomials with two finite critical values. This theorem states that Siegel disks of such polynomials, under a diophantine condition (called Herman's condition) on the rotation number, must have a critical point on their boundaries.Comment: 28 pages. The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-016-2614-

Stiff connections in pseudo-Euclidean manifolds

For a smooth manifold endowed with a (similarity) pseudo-Euclidean structure, a stiff connection $\nabla$ is a symmetric affine connection such that geodesics of $\nabla$ are straight lines of the pseudo-Euclidean structure while the first-order infinitesimal holonomy at each point is an infinitesimal isometry. In this paper, we give a complete classification of stiff connections in a local chart, identify canonical models and start investigating the global geometry of (similarity) pseudo-Euclidean manifolds endowed with a stiff connection. In the conformal class of the pseudo-Euclidean metric g, a stiff connection $\nabla$ defines a pseudo-Riemannian metric h such that unparameterized geodesics of $\nabla$ coincide with unparameterized geodesics of g but have a constant speed with respect to the so-called isochrone metric h. In particular, we obtain a new natural connection on the open unit ball that provides a compromise between properties of Cayley-Klein and Poincar\'e hyperbolic models.Comment: 51 pages, 5 figures, 1 tabl

Similarity surfaces, connections, and the measurable Riemann mapping theorem

This article studies a particular process that approximates solutions of the Beltrami equation (straightening of ellipse fields, a.k.a. measurable Riemann mapping theorem) on $\mathbb{C}$. It passes through the introduction of a sequence of similarity surfaces constructed by gluing polygons, and we explain the relation between their conformal uniformization and the Schwarz-Christoffel formula. Numerical aspects, in particular the efficiency of the process, are not studied, but we draw interesting theoretical consequences. First, we give an independent proof of the analytic dependence, on the data (the Beltrami form), of the solution of the Beltrami equation (Ahlfors-Bers theorem). For this we prove, without using the Ahlfors-Bers theorem, the holomorphic dependence, with respect to the polygons, of the Christoffel symbol appearing in the Schwarz-Christoffel formula. Second, these Christoffel symbols define a sequence of parallel transports on the range, and in the case of a data that is $C^2$ with compact support, we prove that it converges to the parallel transport associated to a particular affine connection, which we identify.Comment: 95 pages, 17 figure

Sur l'implosion parabolique, la taille des disques de Siegel et une conjecture de Marmi, Moussa et Yoccoz

Tout le contenu de ce mémoire est un travail en commun de l'auteur et de Xavier Buff.Pour theta nombre de Brjuno, soit r(theta) le rayon conforme du disque de Siegel de P_theta(z)=exp(i.2.pi.theta)z+z^2 et Phi(theta) la variante due à Yoccoz de la somme de Brjuno. Soit Upsilon(theta) = log r(theta) + Phi(theta). Nous avons démontré précédemment que Upsilon possède un prolongement continu à R, et donné une formule explicite pour sa valeur aux rationnels.La conjecture de Marmi-Moussa-Yoccoz, toujours ouverte, est que la fonction Upsilon est 1/2-Höldérienne.Nous démontrons ici que l'exposant ne peut être amélioré : quel que soit l'intervalle I non vide, Upsilon n'est delta-Höldérienne sur I pour aucun delta>1/2. Sa variation sur I est également non bornée.La preuve est basée sur un développement asymptotique en tout p/q de Upsilon(x_n) pour certaines suites de rationnels x_n tendant vers p/q.L'étude d'un point parabolique et de ses perturbations se fait parfois par l'introduction d'un champ de vecteurs auquel la dynamique est comparée.Nous introduisons un champ de vecteurs particulier qui permet d'une part de donner des estimations suffisamment fines pour effectuer le développement asymptotique de Upsilon(x_n) ; d'autre part de proposer une normalisation intéressante des coordonnées de Fatou d'un point parabolique, dont nous donnons quelques propriétés de base.J'ai apporté un soin particulier à la rédaction de l'implosion parabolique, qu'il a fallu raffiner légèrement et adapter à notre champs de vecteurs