# Noninvertible minimal maps

Sergiĭ Kolyada; L'ubomír Snoha; Sergeĭ Trofimchuk

Fundamenta Mathematicae (2001)

- Volume: 168, Issue: 2, page 141-163
- ISSN: 0016-2736

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topSergiĭ 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 -

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