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Some remarks on the Poincaré-Birkhoff theorem. (English) Zbl 1194.37069
This is a new development on the Poincaré-Birkhoff fixed point theorem. The authors introduce the notion of a ‘positive path’, defined with respect to a homeomorphism of a topological space. This is an oriented continuous curve with the property that the homeomorphism does not map any point of the curve to a preceding point. Using this construct, the authors provide an alternative proof of the Poincaré-Birkhoff theorem (rather, of a generalisation of it due to P. H. Carter). They also give a succinct but well-informed account of the history of the theorem, and connect their work to that of J. Franks.

37E30Homeomorphisms and diffeomorphisms of planes and surfaces
37E40Twist maps
37J10Symplectic mappings, fixed points (dynamical systems)