# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Noninvertible minimal maps

### Tom 168 / 2001

Fundamenta Mathematicae 168 (2001), 141-163 MSC: 37B05, 54H20. DOI: 10.4064/fm168-2-5

#### Streszczenie

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.

#### Autorzy

Tereshchenkivs'ka 3
252601 Kiev, Ukraine
e-mail
Faculty of Natural Sciences
Matej Bel University
Tajovského 40
974 01 Banská Bystrica, Slovakia
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• Sergeĭ TrofimchukDepartamento de Matemáticas