Rotation sets of invariant separating continua of annular homeomorphisms

Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset \mathrm{I}\mathrm{n}\mathrm{t}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_{+}$ and the lower domain $U_{-}$. Fixing a lift of $f$ to the universal cover of $A$, one defines the rotation set $\tilde{\rho }(X)$ of $X$ by means of the invariant probabilities on $X$. For any rational number $p\mathrm{/}q\in \tilde{\rho }(X)$, $f$ is shown to admit a $p\mathrm{/}q$ periodic point in $X$, provided that (1) $X$ consists of nonwandering points or (2) $X$ is an attractor and the frontiers of $U_{-}$ and $U_{+}$ coincides with $X$. Also the Carath'eodory rotation numbers of $U_{\pm }$ are shown to be in $\tilde{\rho }(X)$ for any separating invariant continuum $X$.