W. Thurston constructed a combinatorial model of the Mandelbrot set such that there is a continuous and monotone projection of to this model. We propose the following related model for the space of critically marked cubic polynomials with connected Julia set and all cycles repelling. If , then every point z in the Julia set of the polynomial P defines a unique maximal finite set of angles on the circle corresponding to the rays, whose impressions form a continuum containing z. Let denote the convex hull of . The convex sets partition the closed unit disk. For let be the co-critical point of . We tag the marked dendritic polynomial with the set . Tags are pairwise disjoint; denote by their collection, equipped with the quotient topology. We show that tagging defines a continuous map from to so that serves as a model for .
W. Thurston a construit un modèle combinatoire de l'ensemble de Mandelbrot tel qu'il y ait une projection monotone et continue de sur ce modèle. En relation avec ceci, nous proposons le modèle lié suivant pour l'espace des polynômes cubiques à points critiques marqués, avec ensemble de Julia connexe et tous les cycles répulsifs. Si , alors chaque point z dans l'ensemble de Julia du polynôme P définit un unique ensemble fini maximal d'angles sur le cercle correspondant aux rayons, dont les impressions forment un continuum contenant z. Soit l'enveloppe convexe de . Les ensembles convexes définissent une partition du disque unité fermé. Pour , soit le point co-critique de . Nous balisons le polynôme dendritique marqué avec l'ensemble . Les balises sont deux à deux disjointes ; désignons par leur collection, équipée de la topologie quotient. Nous montrons que le balisage définit une application continue de dans de sorte que est un modèle pour .
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Alexander Blokh 1; Lex Oversteegen 1; Ross Ptacek 2; Vladlen Timorin 2
@article{CRMATH_2017__355_5_590_0, author = {Alexander Blokh and Lex Oversteegen and Ross Ptacek and Vladlen Timorin}, title = {Combinatorial models for spaces of cubic polynomials}, journal = {Comptes Rendus. Math\'ematique}, pages = {590--595}, publisher = {Elsevier}, volume = {355}, number = {5}, year = {2017}, doi = {10.1016/j.crma.2017.04.005}, language = {en}, }
TY - JOUR AU - Alexander Blokh AU - Lex Oversteegen AU - Ross Ptacek AU - Vladlen Timorin TI - Combinatorial models for spaces of cubic polynomials JO - Comptes Rendus. Mathématique PY - 2017 SP - 590 EP - 595 VL - 355 IS - 5 PB - Elsevier DO - 10.1016/j.crma.2017.04.005 LA - en ID - CRMATH_2017__355_5_590_0 ER -
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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