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 ⊆ X then both f(A) and 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm168-2-5,
author = {Sergi\u\i\ Kolyada and L'ubom\'\i r Snoha and Serge\u\i\ Trofimchuk},
title = {Noninvertible minimal maps},
journal = {Fundamenta Mathematicae},
volume = {167},
year = {2001},
pages = {141-163},
zbl = {1031.37014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm168-2-5}
}
Sergiĭ Kolyada; L'ubomír Snoha; Sergeĭ Trofimchuk. Noninvertible minimal maps. Fundamenta Mathematicae, Tome 167 (2001) pp. 141-163. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm168-2-5/