Soit une surface compacte connexe et soit un élément du groupe des homéomorphismes de isotopes à l’identité. Notons un relevé de au revêtement universel de . Fixons un domaine fondamental de ce revêtement universel. On dit que l’homéomorphisme est non-diffus si la suite converge vers , où désigne le diamètre de . Supposons que la surface est orientable de bord non vide. Nous démontrons que, si est distincte du disque et de l’anneau, un homéomorphisme est non-diffus si et seulement si il a des conjugués dans arbitrairement proches de l’identité. Dans le cas où la surface est l’anneau, nous démontrons qu’un homéomorphisme est non-diffus si et seulement si il a des conjugués arbitrairement proches d’une rotation (ce résultat était déjà connu dans la plupart des cas grâce à un théorème dû à Béguin, Crovisier, Le Roux et Patou). On en déduit que, pour de telles surfaces , un élément de est distordu si et seulement si il est non diffus.
Let be a compact connected surface and let be an element of the group of homeomorphisms of isotopic to the identity. Denote by a lift of to the universal cover of . Fix a fundamental domain of this universal cover. The homeomorphism is said to be non-spreading if the sequence converges to , where is the diameter of . Let us suppose now that the surface is orientable with a nonempty boundary. We prove that, if is different from the annulus and from the disc, a homeomorphism is non-spreading if and only if it has conjugates in arbitrarily close to the identity. In the case where the surface is the annulus, we prove that a homeomorphism is non-spreading if and only if it has conjugates in arbitrarily close to a rotation (this was already known in most cases by a theorem by Béguin, Crovisier, Le Roux and Patou). We deduce that, for such surfaces , an element of is distorted if and only if it is non-spreading.
Accepté le :
Publié le :
DOI : 10.5802/jep.78
Keywords: Topological dynamics, homeomorphism, distortion, conjugacy class
Mot clés : Dynamique topologique, homéomorphisme, distorsion, classe de conjugaison
@article{JEP_2018__5__565_0, author = {Militon, Emmanuel}, title = {Conjugacy class of homeomorphisms and distortion elements in groups~of~homeomorphisms}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {565--604}, publisher = {Ecole polytechnique}, volume = {5}, year = {2018}, doi = {10.5802/jep.78}, mrnumber = {3852261}, zbl = {06988588}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jep.78/} }
TY - JOUR AU - Militon, Emmanuel TI - Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms JO - Journal de l’École polytechnique — Mathématiques PY - 2018 SP - 565 EP - 604 VL - 5 PB - Ecole polytechnique UR - http://archive.numdam.org/articles/10.5802/jep.78/ DO - 10.5802/jep.78 LA - en ID - JEP_2018__5__565_0 ER -
%0 Journal Article %A Militon, Emmanuel %T Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms %J Journal de l’École polytechnique — Mathématiques %D 2018 %P 565-604 %V 5 %I Ecole polytechnique %U http://archive.numdam.org/articles/10.5802/jep.78/ %R 10.5802/jep.78 %G en %F JEP_2018__5__565_0
Militon, Emmanuel. Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 565-604. doi : 10.5802/jep.78. http://archive.numdam.org/articles/10.5802/jep.78/
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