INSTITUTE OF MATHEMATICS · POLISH ACADEMY OF SCIENCES
FUNDAMENTA
MATHEMATICAE
ISSN: 0016-2736(p) 1730-6329(e)
Noninvertible minimal maps
Sergiĭ Kolyada1, L'ubomír Snoha2, Sergeĭ Trofimchuk3 Fund. Math. 168 (2001), 141-163
doi:10.4064/fm168-2-5 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.
