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Proper minimal sets on compact connected 2-manifolds are nowhere dense

Published online by Cambridge University Press:  01 June 2008

SERGII˘ KOLYADA,
L’UBOMÍR SNOHA and
SERGEI˘ TROFIMCHUK
SERGII˘ KOLYADA
Affiliation:
Institute of Mathematics, NASU, Tereshchenkivs’ka 3, 01601 Kiev, Ukraine (email: skolyada@imath.kiev.ua)
L’UBOMÍR SNOHA
Affiliation:
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia (email: snoha@fpv.umb.sk)
SERGEI˘ TROFIMCHUK
Affiliation:
Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile (email: trofimch@inst-mat.utalca.cl)
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Abstract

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Let $\mathcal {M}^2$ be a compact connected two-dimensional manifold, with or without boundary, and let $f:{\mathcal {M}}^2\to \mathcal {M}^2$ be a continuous map. We prove that if $M \subseteq \mathcal {M}^2$ is a minimal set of the dynamical system $(\mathcal {M}^2,f)$ then either $M = \mathcal {M}^2$ or M is a nowhere dense subset of $\mathcal {M}^2$. Moreover, we add a shorter proof of the recent result of Blokh, Oversteegen and Tymchatyn, that in the former case $\mathcal {M}^2$ is a torus or a Klein bottle.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 3 , June 2008, pp. 863 - 876
Copyright
Copyright © Cambridge University Press 2008

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