Topology/Dynamical Systems
Minimal sets of R-closed surface homeomorphisms
Comptes Rendus. Mathématique, Volume 350 (2012) no. 23-24, pp. 1051-1053.

Let M be an orientable connected closed surface and f be a homeomorphism on M. Suppose that the set $Eˆf:={(x,y)|y∈Of(x)¯}$ 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 $Eˆf$ of a homeomorphism f on a compact metrizable space is closed, then so is $Eˆfk$ for any natural number k.

Soit M une surface fermée connexe orientable sans bord et f un homéomorphisme de M. Supposons que lʼensemble $Eˆf:={f(x,y)|y∈Of(x)¯}$ 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 $Eˆf$ dʼun homéomorphisme f dʼun espace compact métrisable est fermé, alors les $Eˆfk$ sont fermés pour tout entier naturel k.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.11.002

Tomoo Yokoyama 1

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