Noninvertible minimal maps

Sergiĭ Kolyada ¹   ; L'ubomír Snoha ²   ; Sergeĭ Trofimchuk ³

¹ Institute of Mathematics Ukrainian Academy of Sciences Tereshchenkivs'ka 3 252601 Kiev, Ukraine
² Department of Mathematics Faculty of Natural Sciences Matej Bel University Tajovského 40 974 01 Banská Bystrica, Slovakia
³ Departamento de Matemáticas Facultad de Ciencias Universidad de Chile Las Palmeras 3425 Santiago, Chile

Fundamenta Mathematicae, Tome 168 (2001) no. 2, pp. 141-163

Résumé

For a discrete dynamical system given by a compact Hausdorff space $X$ and a continuous selfmap $f$ of $X$ the connection between minimality, invertibility and openness of $f$ is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if $f$ is minimal and $A\subseteq X$ then both $f(A)$ and $f^{-1}(A)$ share with $A$ those topological properties which describe how large a set is. Using these results it is proved that any minimal map in a compact metric space is almost one-to-one and, moreover, when restricted to a suitable invariant residual set it becomes a minimal homeomorphism. Finally, two kinds of examples of noninvertible minimal maps on the torus are given—these are obtained either as a factor or as an extension of an appropriate minimal homeomorphism of the torus.

Détail

DOI : 10.4064/fm168-2-5

Keywords: discrete dynamical system given compact hausdorff space continuous selfmap connection between minimality invertibility openness investigated shown minimal map feebly sends sets sets nonempty interiors homeomorphism further shown minimal subseteq share those topological properties which describe large set using these results proved minimal map compact metric space almost one to one moreover restricted suitable invariant residual set becomes minimal homeomorphism finally kinds examples noninvertible minimal maps torus given these obtained either factor extension appropriate minimal homeomorphism torus