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Noninvertible minimal maps

Sergiĭ KolyadaL'ubomír SnohaSergeĭ Trofimchuk — 2001

Fundamenta Mathematicae

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...

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