Combinatorial models for spaces of cubic polynomials

Alexander Blokh; Lex Oversteegen; Ross Ptacek; Vladlen Timorin

doi:10.1016/j.crma.2017.04.005

Dynamical systems

Combinatorial models for spaces of cubic polynomials

Presented by: the Editorial Board

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

Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 590-595.

Abstract
Résumé

W. Thurston constructed a combinatorial model of the Mandelbrot set $M_{2}$ such that there is a continuous and monotone projection of $M_{2}$ to this model. We propose the following related model for the space ${MD}_{3}$ of critically marked cubic polynomials with connected Julia set and all cycles repelling. If $(P, c_{1}, c_{2}) \in {MD}_{3}$ , then every point z in the Julia set of the polynomial P defines a unique maximal finite set $A_{z}$ of angles on the circle corresponding to the rays, whose impressions form a continuum containing z. Let $G (z)$ denote the convex hull of $A_{z}$ . The convex sets $G (z)$ partition the closed unit disk. For $(P, c_{1}, c_{2}) \in {MD}_{3}$ let $c_{1}^{⁎}$ be the co-critical point of $c_{1}$ . We tag the marked dendritic polynomial $(P, c_{1}, c_{2})$ with the set $G (c_{1}^{⁎}) \times G (P (c_{2})) \subset \overline{D} \times \overline{D}$ . Tags are pairwise disjoint; denote by ${MD}_{3}^{comb}$ their collection, equipped with the quotient topology. We show that tagging defines a continuous map from ${MD}_{3}$ to ${MD}_{3}^{comb}$ so that ${MD}_{3}^{comb}$ serves as a model for ${MD}_{3}$ .

W. Thurston a construit un modèle combinatoire de l'ensemble de Mandelbrot $M_{2}$ tel qu'il y ait une projection monotone et continue de $M_{2}$ sur ce modèle. En relation avec ceci, nous proposons le modèle lié suivant pour l'espace ${MD}_{3}$ des polynômes cubiques à points critiques marqués, avec ensemble de Julia connexe et tous les cycles répulsifs. Si $(P, c_{1}, c_{2}) \in {MD}_{3}$ , alors chaque point z dans l'ensemble de Julia du polynôme P définit un unique ensemble fini maximal $A_{z}$ d'angles sur le cercle correspondant aux rayons, dont les impressions forment un continuum contenant z. Soit $G (z)$ l'enveloppe convexe de $A_{z}$ . Les ensembles convexes $G (z)$ définissent une partition du disque unité fermé. Pour $(P, c_{1}, c_{2}) \in {MD}_{3}$ , soit $c_{1}^{⁎}$ le point co-critique de $c_{1}$ . Nous balisons le polynôme dendritique marqué $(P, c_{1}, c_{2})$ avec l'ensemble $G (c_{1}^{⁎}) \times G (P (c_{2})) \subset \overline{D} \times \overline{D}$ . Les balises sont deux à deux disjointes ; désignons par ${MD}_{3}^{comb}$ leur collection, équipée de la topologie quotient. Nous montrons que le balisage définit une application continue de ${MD}_{3}$ dans ${MD}_{3}^{comb}$ de sorte que ${MD}_{3}^{comb}$ est un modèle pour ${MD}_{3}$ .

Received: 2014-09-19
Accepted: 2017-04-07
Published online: 2017-04-20

DOI: 10.1016/j.crma.2017.04.005

Author's affiliations:

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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     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},
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     number = {5},
     year = {2017},
     doi = {10.1016/j.crma.2017.04.005},
     language = {en},
}

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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/

References
Cited by

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[2] A. Blokh; L. Oversteegen; R. Ptacek; V. Timorin Laminational models for spaces of polynomials of any degree, 2014 (preprint second version 2016) | arXiv

[3] A. Blokh; L. Oversteegen; R. Ptacek; V. Timorin Models for spaces of dendritic polynomials, 2017 (preprint) | arXiv

[4] L. Goldberg; J. Milnor Fixed points of polynomial maps. Part II. Fixed point portraits, Ann. Sci. Éc. Norm. Supér. (4), Volume 26 (1993), pp. 51-98

[5] J. Kiwi $R$ eal laminations and the topological dynamics of complex polynomials, Adv. Math., Volume 184 (2004), pp. 207-267

[6] D. Schleicher On fibers and local connectivity of Mandelbrot and multibrot sets (M. Lapidus; M. van Frankenhuysen, eds.), Fractal Geometry and Applications: a Jubilee of Benoit Mandelbrot, Proc. Symp. Pure Math., vol. 72, American Mathematical Society, Providence, RI, USA, 2004, pp. 477-517 (Part 1)

[7] W. Thurston The combinatorics of iterated rational maps (1985) (D. Schleicher; A.K. Peters, eds.), Complex Dynamics: Families and Friends, 2009, pp. 1-108

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