Noninvertible minimal maps

Sergii Kolyada, Lubomir Snoha and Sergei Trofimchuk

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 even 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 those topological properties with \$A\$ 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.