Some extensions of the Poincaré–Birkhoff theorem to the cylinder and a remark on mappings of the torus homotopic to Dehn twists

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-preserving case, the subset of diffeomorphisms of the torus which have a periodic orbit is dense in the set of diffeomorphisms of the torus in any topology.Finally, we extend some theorems we already obtained for twist mappings to a more general setting.