We propose, in the general setting of topological spaces, a definition of two-dimensional
oriented cell and consider maps which possess a property of stretching along the paths
with respect to oriented cells. For these maps, we prove some theorems on the existence
of fixed points, periodic points, and sequences of iterates which are chaotic in a suitable
manner. Our results, motivated by the study of the Poincaré map associated to some nonlinear Hill’s equations, extend and improve some recent work. The proofs are elementary
in the sense that only well-known properties of planar sets and maps and a
two-dimensional equivalent version of the Brouwer fixed point theorem are used.
