Prime end rotation numbers of invariant separating contunua 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 $X$, as well as the prime end rotation number ${\check{\rho }}_{\pm }$ of $U_{\pm }$. The purpose of this paper is to show that ${\check{\rho }}_{\pm }$ belongs to $\tilde{\rho }(X)$ for any separating invariant continuum $X$.