Noninvertible minimal maps

Tom 168 / 2001

Sergiĭ Kolyada, L'ubomír Snoha, Sergeĭ Trofimchuk Fundamenta Mathematicae 168(2001), 141-163 MSC: 37B05, 54H20. DOI: 10.4064/fm168-2-5


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 $A\subseteq X$ 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.


  • Sergiĭ KolyadaInstitute of Mathematics
    Ukrainian Academy of Sciences
    Tereshchenkivs'ka 3
    252601 Kiev, Ukraine
  • L'ubomír SnohaDepartment of Mathematics
    Faculty of Natural Sciences
    Matej Bel University
    Tajovského 40
    974 01 Banská Bystrica, Slovakia
  • Sergeĭ TrofimchukDepartamento de Matemáticas
    Facultad de Ciencias
    Universidad de Chile
    Las Palmeras 3425
    Santiago, Chile

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