Fixed points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells

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.