  It is well-known that when a positively expansive dynamical system is
invertible then its underlying space is finite. C.Morales has introduced a
decade ago a natural way to generalize positive expansiveness, by introducing
other properties that he called positive $n$-expansiveness, for all $n \ge 1$,
positive $1$-expansiveness being identical to positive expansiveness. Contrary
to positive expansiveness, positive $n$-expansiveness for $n>1$ does not
enforce that the space is finite when the system is invertible. In the present
paper we call finitely positively expansive dynamical systems as the ones which
are positively $n$-expansive for some integer $n$, and prove several results on
this class of systems. In particular, the well-known result quoted above is
true if we add the constraint of shadowing property, while it is not if this
property is replaced with minimality. Furthermore, finitely positively
expansive systems cannot occur on certain topological spaces such as the
interval, when the system is assumed to be invertible finite positive
expansiveness implies zero topological entropy. Overall we show that the class
of finitely positively expansive dynamical systems is quite rich and leave
several questions open for further research.
