Some remarks on the Poincaré-Birkhoff theorem. (English) Zbl 1194.37069

Le Calvez, Patrice; Wang, Jian

Proc. Am. Math. Soc. 138, No. 2, 703-715 (2010).

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.

Reviewer: Franco Vivaldi (London)

MSC:

37E30	Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37E40	Dynamical aspects of twist maps
37J10	Symplectic mappings, fixed points (dynamical systems) (MSC2010)

Keywords:

fixed points; boundary twist condition; positive path

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