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 $X\subset {\rm Int}A$ 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 $p/q\in \tilde\rho(X)$, $f$ is shown to admit a $p/q$ 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 $X$.