We prove that if $F$ is an orientation-preserving homeomorphism of the plane
that leaves invariant an irreducible plane separating continuum $\Delta$,
then, with the possible exception of three numbers, if $p/q$ is a reduced
rational in the interior of the convex hull of the rotation set of
$F\vert_{\Delta}$ (with respect to some lift) there are at least two distinct
periodic orbits of $F\vert_{\Delta}$ of period $q$ and rotation number $p/q$.
This result also applies to certain nonseparating invariant
continua.