Comptes Rendus
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)|yOf(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)|yOf(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.

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
     author = {Tomoo Yokoyama},
     title = {Minimal sets of {\protect\emph{R}-closed} surface homeomorphisms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1051--1053},
     publisher = {Elsevier},
     volume = {350},
     number = {23-24},
     year = {2012},
     doi = {10.1016/j.crma.2012.11.002},
     language = {en},
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JO  - Comptes Rendus. Mathématique
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VL  - 350
IS  - 23-24
PB  - Elsevier
DO  - 10.1016/j.crma.2012.11.002
LA  - en
ID  - CRMATH_2012__350_23-24_1051_0
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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.

[1] A. Biś; H. Nakayama; P. Walczak Locally connected exceptional minimal sets of surface homeomorphisms, Ann. Inst. Fourier, Volume 54 (2004), pp. 711-732

[2] P. Erdös; A.H. Stone Some remarks on almost periodic transformations, Bull. Amer. Math. Soc., Volume 51 (1945), pp. 126-130

[3] W. Gottschalk; G. Hedlund Topological Dynamics, Amer. Math. Soc. Publ., vol. 36, Amer. Math. Soc., Providence, RI, 1955

[4] T. Jäger; F. Kwakkel; A. Passeggi A classification of minimal sets of torus homeomorphisms, Math. Z. (2012) | arXiv | DOI

[5] F. Kwakkel Minimal sets of non-resonant torus homeomorphisms, Fund. Math., Volume 211 (2011), pp. 41-76

[6] D. Montgomery Pointwise periodic homeomorphisms, Amer. J. Math., Volume 59 (1937), pp. 118-120

[7] T. Yokoyama Recurrence, pointwise almost periodicity and orbit closure relation for flows and foliations | arXiv

[8] T. Yokoyama R-closed homeomorphisms on surfaces | arXiv

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