Some remarks on the Poincaré-Birkhoff theorem

Abstract

We define the notion of a positive path of a homeomorphism of a topological space. It seems to be a natural object to understand Birkhoff’s arguments in his proof of the celebrated Poincaré-Birkhoff theorem. We write the proof of this theorem, by using positive paths, and the proof of its generalization due to P. Carter. We will also explain the links with the free disk chains introduced in the subject by J. Franks. We will finish the paper by studying the local versions where the upper curve is not invariant and will explain why this curve or its image must be a graph to get such a generalization.