On torus homeomorphisms whose rotation set is an interval

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We prove that for a homeomorphism $$\tilde{f}:\mathbf T ^2\rightarrow \mathbf T ^2$$ in the homotopy class of the identity and with a lift $$f:\mathbf R ^2\rightarrow \mathbf R ^2$$ whose rotation set $$\rho (f)$$ is an interval, either every rational point in $$\rho (f)$$ is realized by a periodic orbit, or the dynamics of $$\tilde{f}$$ is annular, in the sense that there exists a periodic, essential, annular set for $$\tilde{f}$$ . In the latter case we also give a qualitative description of the dynamics.