Noninvertible minimal maps

Sergiĭ Kolyada, L'ubomír Snoha, and Sergeĭ Trofimchuk. "Noninvertible minimal maps." Fundamenta Mathematicae 168.2 (2001): 141-163. <http://eudml.org/doc/281909>.

@article{SergiĭKolyada2001,
abstract = {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 $f^\{-1\}(A)$ 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.},
author = {Sergiĭ Kolyada, L'ubomír Snoha, Sergeĭ Trofimchuk},
journal = {Fundamenta Mathematicae},
keywords = {minimal dynamical system; noninvertible map; irreducible map; open map; feebly open map; factor; extension; almost one-to-one map; torus},
language = {eng},
number = {2},
pages = {141-163},
title = {Noninvertible minimal maps},
url = {http://eudml.org/doc/281909},
volume = {168},
year = {2001},
}

TY - JOUR
AU - Sergiĭ Kolyada
AU - L'ubomír Snoha
AU - Sergeĭ Trofimchuk
TI - Noninvertible minimal maps
JO - Fundamenta Mathematicae
PY - 2001
VL - 168
IS - 2
SP - 141
EP - 163
AB - 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 $f^{-1}(A)$ 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.
LA - eng
KW - minimal dynamical system; noninvertible map; irreducible map; open map; feebly open map; factor; extension; almost one-to-one map; torus
UR - http://eudml.org/doc/281909
ER -