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-

On (non-)local-connectivity of some Julia sets

This article deals with the question of local connectivity of the Julia set of polynomials and rational maps. It essentially presents conjectures and questions.Comment: 28 pages, 3 figure

Rigidity of non-renormalizable Newton maps

Non-renormalizable Newton maps are rigid. More precisely, we prove that their Julia set carries no invariant line fields and that the topological conjugacy is equivalent to quasi-conformal conjugacy in this case

Newton maps as matings of cubic polynomials

International audienceIn this paper we prove existence and uniqueness of matings of a large class of renormalizable cubic polynomials with one fixed critical point and the other cubic polynomial having two fixed critical points. The resulting mating is a Newton map. Our result is the first part towards a conjecture by Tan Lei, stating that all (cubic) Newton maps can be described as matings or captures

Carrots for dessert

Carrots for dessert is the title of a section of the paper `On polynomial-like mappings' by Douady and Hubbard. In that section the authors define a notion of dyadic carrot fields of the Mandelbrot set M and more generally for Mandelbrot like families. They remark that such carrots are small when the dyadic denominator is large, but they do not even try to prove a precise such statement. In this paper we formulate and prove a precise statement of asymptotic shrinking of dyadic Carrot-fields around M. The same proof carries readily over to show that the dyadic decorations of copies M' of the Mandelbrot set M inside M and inside the parabolic Mandelbrot set shrink to points when the denominator diverge to infinity.Comment: 21 pages, 2 figure

Hyperbolic components of polynomials with a fixed critical point of maximal order

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