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In this paper we prove some extensions of the Poincaré-Birkhoff theorem to S 1 × R. We consider mappings h of the cylinder which satisfy an 'infinity twist condition' and prove that under certain additional hypotheses, for every rational p/q, h has q-periodic orbits with rotation number p/q. And in many interesting cases these orbits have non-null topological indices so they appear at least in pairs. We also obtain, as a consequence of some of the above results, that in the area-preservingdoi:10.1088/0951-7715/18/5/018 fatcat:qnp46r4drjcstjzkw3ivhrccoy