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.