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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.

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