Comptes Rendus
Dynamical systems
Combinatorial models for spaces of cubic polynomials
Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 590-595.

W. Thurston constructed a combinatorial model of the Mandelbrot set M2 such that there is a continuous and monotone projection of M2 to this model. We propose the following related model for the space MD3 of critically marked cubic polynomials with connected Julia set and all cycles repelling. If (P,c1,c2)MD3, then every point z in the Julia set of the polynomial P defines a unique maximal finite set Az of angles on the circle corresponding to the rays, whose impressions form a continuum containing z. Let G(z) denote the convex hull of Az. The convex sets G(z) partition the closed unit disk. For (P,c1,c2)MD3 let c1 be the co-critical point of c1. We tag the marked dendritic polynomial (P,c1,c2) with the set G(c1)×G(P(c2))D×D. Tags are pairwise disjoint; denote by MD3comb their collection, equipped with the quotient topology. We show that tagging defines a continuous map from MD3 to MD3comb so that MD3comb serves as a model for MD3.

W. Thurston a construit un modèle combinatoire de l'ensemble de Mandelbrot M2 tel qu'il y ait une projection monotone et continue de M2 sur ce modèle. En relation avec ceci, nous proposons le modèle lié suivant pour l'espace MD3 des polynômes cubiques à points critiques marqués, avec ensemble de Julia connexe et tous les cycles répulsifs. Si (P,c1,c2)MD3, alors chaque point z dans l'ensemble de Julia du polynôme P définit un unique ensemble fini maximal Az d'angles sur le cercle correspondant aux rayons, dont les impressions forment un continuum contenant z. Soit G(z) l'enveloppe convexe de Az. Les ensembles convexes G(z) définissent une partition du disque unité fermé. Pour (P,c1,c2)MD3, soit c1 le point co-critique de c1. Nous balisons le polynôme dendritique marqué (P,c1,c2) avec l'ensemble G(c1)×G(P(c2))D×D. Les balises sont deux à deux disjointes ; désignons par MD3comb leur collection, équipée de la topologie quotient. Nous montrons que le balisage définit une application continue de MD3 dans MD3comb de sorte que MD3comb est un modèle pour MD3.

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DOI: 10.1016/j.crma.2017.04.005

Alexander Blokh 1; Lex Oversteegen 1; Ross Ptacek 2; Vladlen Timorin 2

1 Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA
2 Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva St., 119048 Moscow, Russia
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Alexander Blokh; Lex Oversteegen; Ross Ptacek; Vladlen Timorin. Combinatorial models for spaces of cubic polynomials. Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 590-595. doi : 10.1016/j.crma.2017.04.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.04.005/

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