Minimal sets of <i>R</i>-closed surface homeomorphisms

Tomoo Yokoyama

doi:10.1016/j.crma.2012.11.002

Topology/Dynamical Systems

Minimal sets of R-closed surface homeomorphisms
[Ensembles minimaux des homéomorphismes R-fermés de surfaces]

Présenté par : the Editorial Board

Tomoo Yokoyama ¹

¹ Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, 060-0810, Japan

Comptes Rendus. Mathématique, Volume 350 (2012) no. 23-24, pp. 1051-1053.

Résumé
Abstract

Soit M une surface fermée connexe orientable sans bord et f un homéomorphisme de M. Supposons que lʼensemble ${\hat{E}}_{f} : = {f (x, y) | y \in \bar{O_{f} (x)}}$ soit fermé. Alors la suspension vérifie lʼune des conditions suivantes : 1) lʼadhérence dʼun élément est un tore ; 2) il existe un ensemble minimal qui nʼest pas localement connexe. De plus, on démontre que si lʼensemble ${\hat{E}}_{f}$ dʼun homéomorphisme f dʼun espace compact métrisable est fermé, alors les ${\hat{E}}_{f^{k}}$ sont fermés pour tout entier naturel k.

Let M be an orientable connected closed surface and f be a homeomorphism on M. Suppose that the set ${\hat{E}}_{f} : = {(x, y) | y \in \bar{O_{f} (x)}}$ is closed. Then the suspension of f satisfies one of the following conditions: 1) the closure of each element of it is toral; 2) there is a minimal set which is not locally connected. Moreover, we show that if the set ${\hat{E}}_{f}$ of a homeomorphism f on a compact metrizable space is closed, then so is ${\hat{E}}_{f^{k}}$ for any natural number k.

Reçu le : 2012-08-15
Accepté le : 2012-11-06
Publié le : 2012-11-17

DOI : 10.1016/j.crma.2012.11.002

Affiliations des auteurs :

Tomoo Yokoyama ¹

¹ Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, 060-0810, Japan

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     title = {Minimal sets of {\protect\emph{R}-closed} surface homeomorphisms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1051--1053},
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     doi = {10.1016/j.crma.2012.11.002},
     language = {en},
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Tomoo Yokoyama. Minimal sets of R-closed surface homeomorphisms. Comptes Rendus. Mathématique, Volume 350 (2012) no. 23-24, pp. 1051-1053. doi : 10.1016/j.crma.2012.11.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.11.002/

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