Noninvertible minimal maps
Volume 168 / 2001
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 $A\subseteq X$ 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.
