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.

Received:
Accepted:
Published online:
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
@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  - 
%0 Journal Article
%A Alexander Blokh
%A Lex Oversteegen
%A Ross Ptacek
%A Vladlen Timorin
%T Combinatorial models for spaces of cubic polynomials
%J Comptes Rendus. Mathématique
%D 2017
%P 590-595
%V 355
%N 5
%I Elsevier
%R 10.1016/j.crma.2017.04.005
%G en
%F CRMATH_2017__355_5_590_0
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/

[1] A. Blokh; D. Mimbs; L. Oversteegen; K. Valkenburg Laminations in the language of leaves, Trans. Amer. Math. Soc., Volume 365 (2013), pp. 5367-5391

[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 Real 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

Cited by Sources:

Comments - Policy


Articles of potential interest

Wandering triangles exist

Alexander Blokh; Lex Oversteegen

C. R. Math (2004)