Wandering triangles exist

Alexander Blokh; Lex Oversteegen

doi:10.1016/j.crma.2004.06.024

Dynamical Systems

Wandering triangles exist
[L'existence des triangles errants.]

Présenté par : Jean-Christophe Yoccoz

Alexander Blokh ¹ ; Lex Oversteegen ¹

¹ Department of Mathematics, UAB, Birmingham, AL 35294, USA

Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 365-370.

Abstract (VO)
Résumé

W.P. Thurston introduced closed $σ_{d}$ -invariant laminations (where $σ_{d} = z^{d} : S^{1} \to S^{1}$ , $d ⩾ 2$ ) as a tool in complex dynamics. He defined wandering triangles as triples $T \subset S^{1}$ such that $σ_{d}^{n} (T)$ consists of three distinct points for all $n ⩾ 0$ and the convex hulls of all the sets $σ_{d}^{n} (T)$ in the plane are pairwise disjoint, and proved that $σ_{2}$ admits no wandering triangles. We show that for every $d ⩾ 3$ there exist uncountably many $σ_{d}$ -invariant closed laminations with wandering triangles and pairwise non-conjugate factor maps of $σ_{d}$ on the corresponding quotient spaces.

Les laminations fermées $σ_{d}$ -invariantes (où $σ_{d} = z^{d} : S^{1} \to S^{1}, d ⩾ 2$ ) ont été introduites par W. P. Thurston comme un outil pour l'étude des systèmes dynamiques dans le plan complexe. Il avait défini les triangles errants comme étant des triplets $T \subset S^{1}$ tels que $σ_{d}^{n} (T)$ est composé des trois points distincts pour tout $n ⩾ 0$ , et les enveloppes convexes de tous les ensembles $σ_{d}^{n} (T)$ sont deux-à-deux disjointes dans le plan complexe. Il avait démontré que $σ_{2}$ n'admet pas des triangles errants. Nous montrons que pour tout $d ⩾ 3$ il existe une collection nondénombrable de laminations fermées $σ_{d}$ -invariantes qui ont des triangles errants et des applications-facteurs de $σ_{d}$ non-conjuguées, deux-à-deux distinctes, sur les espaces quotients associés.

Reçu le : 2004-06-17
Accepté le : 2004-06-22
Publié le : 2004-08-10

DOI : 10.1016/j.crma.2004.06.024

Affiliations des auteurs :

Alexander Blokh ¹ ; Lex Oversteegen ¹

¹ Department of Mathematics, UAB, Birmingham, AL 35294, USA

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     title = {Wandering triangles exist},
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Alexander Blokh; Lex Oversteegen. Wandering triangles exist. Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 365-370. doi : 10.1016/j.crma.2004.06.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.06.024/

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Nada Alhabib; Lasse Rempe-Gillen Escaping Endpoints Explode, Computational Methods and Function Theory, Volume 17 (2017) no. 1, p. 65 | DOI:10.1007/s40315-016-0169-8
Alexander Blokh; Doug Childers; Genadi Levin; Lex Oversteegen; Dierk Schleicher An extended Fatou–Shishikura inequality and wandering branch continua for polynomials, Advances in Mathematics, Volume 288 (2016), p. 1121 | DOI:10.1016/j.aim.2015.10.020

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