Noninvertible minimal maps (Q2717753)

Let \(X\) be a compact metric space, and let \(f:X\to X\) be a minimal map, this means a continuous map, such that for each \(x\in X\) the forward orbit of \(x\) is dense. The problem investigated in this paper is the relation between minimality and invertibility. A function is called feebly open, if the image of any nonempty open set has nonempty interior. This is equivalent to the property that the preimage of every dense set is dense. It is proved that every minimal map is feebly open. Therefore for a minimal map \(f\) the following properties are equivalent: i) \(f\) is open, ii) \(f\) is invertible, iii) \(f\) is a homeomorphism. Then the authors prove that every minimal map \(f\) is almost one-to-one, this means that the set of all points having exactly one preimage is a dense \(G_{\delta}\). From this result it is deduced that there exists a (in general noncompact) residual set \(Y\subseteq X\) such that \(f|_{Y}\) and \((f|_{Y})^{-1}\) are minimal homeomorphisms. Obviously \(f|_{Y}\) is uniformly continuous. On the other hand \((f|_{Y})^{-1}\) is uniformly continuous, if and only if \(f\) is a homeomorphism. Conditions on an invertible minimal map \(h\) on a noncompact set \(Y\) are given, which imply that \(h\) can be extended to a minimal (noninvertible) map on the completion of \(Y\).NEWLINENEWLINENEWLINEThen a noninvertible minimal map on the (two-dimensional) torus is constructed. This shows that on the torus there exist both minimal homeomorphisms and minimal noninvertible maps. For this example of a noninvertible minimal map the authors use a construction by \textit{M. Rees} [Isr. J. Math. 32, 201-208 (1979; Zbl 0403.54036)]. The desired map can be obtained as a factor of Rees' example. It is also possible to obtain this map by modifying Rees' construction. In an appendix the authors give all details of this construction.NEWLINENEWLINENEWLINEThe paper is very well written and therefore easy to read. This applies also to the appendix, although it is only a small variation of Rees' construction. Since some misprints make Rees' original paper hard to read, the reader finds here an easier readable description.