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Noninvertible minimal maps

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Kolyada S., Snoha L. u., Trofimchuk S.

Fundamenta Mathematicae

2001

tom 168

nr 2

141-163

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.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Kolyada S.

1 Snoha L. u.

1 Trofimchuk S.

1 2001

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Noninvertible minimal maps