Rotation sets of invariant separating continua of annular homeomorphisms

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Let f be a homeomorphism of the closed annulus A isotopic to the identity, and let X⊂ IntA 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 ρ(X) of X by means of the invariant probabilities on X. For any rational number p/q∈ ρ(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 Carathéodory rotation numbers of U± are shown to be in ρ(X) for any separating invariant continuum X.