[Groupes d’homéomorphismes d’ensembles de Julia : l’avion, la basilique et les lapins]
The airplane, the basilica and the Douady rabbit (and, more generally, rabbits with more than two ears) are well-known Julia sets of complex quadratic polynomials. In this paper we study the groups of all homeomorphisms of such fractals and of all automorphisms of their laminations. In particular, we identify them with some kaleidoscopic groups or universal groups and thus realize them as Polish permutation groups. From these identifications, we deduce algebraic, topological and geometric properties of these groups.
L’avion, la basilique et le lapin de Douady (et, plus généralement, les lapins à plus de deux oreilles) sont des ensembles de Julia bien connus de polynômes quadratiques complexes. Dans cet article, nous étudions les groupes de tous les homéomorphismes de ces fractales et de tous les automorphismes de leurs laminations. En particulier, nous les identifions à certains groupes kaléidoscopiques ou groupes universels et les réalisons ainsi comme des groupes de permutation polonais. À partir de ces identifications, nous déduisons les propriétés algébriques, topologiques et géométriques de ces groupes.
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Keywords: Julia sets, homeomorphisms groups, topological dynamics, Polish groups
Mots-clés : Ensembles de Julia, groupes d’homéomorphismes, dynamique topologique, groupes polonais
Bruno Duchesne 1 ; Matteo Tarocchi 1
CC-BY 4.0
@article{JEP_2026__13__255_0,
author = {Bruno Duchesne and Matteo Tarocchi},
title = {Homeomorphism groups of basilica, rabbit and airplane {Julia} sets},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {255--320},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.327},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.327/}
}
TY - JOUR AU - Bruno Duchesne AU - Matteo Tarocchi TI - Homeomorphism groups of basilica, rabbit and airplane Julia sets JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 255 EP - 320 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.327/ DO - 10.5802/jep.327 LA - en ID - JEP_2026__13__255_0 ER -
%0 Journal Article %A Bruno Duchesne %A Matteo Tarocchi %T Homeomorphism groups of basilica, rabbit and airplane Julia sets %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 255-320 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.327/ %R 10.5802/jep.327 %G en %F JEP_2026__13__255_0
Bruno Duchesne; Matteo Tarocchi. Homeomorphism groups of basilica, rabbit and airplane Julia sets. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 255-320. doi: 10.5802/jep.327
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