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Zbl 0676.58037
Franks, John
Generalizations of the Poincaré-Birkhoff theorem.
(English)
[J] Ann. Math. (2) 128, No.1, 139-151 (1988); erratum 164, 1097-1098 (2006) ISSN 0003-486X; ISSN 1939-0980/e

The author proves some generalizations of the classical Poincaré- Birkhoff theorem on area-preserving homeomorphisms of the annulus which satisfy a boundary twist condition. Briefly speaking, in his generalization, the existence of positively returning disks and negatively returning disks replaces the boundary twist condition. Furthermore, the area-preserving hypothesis is replaced by the weaker condition that every point be non-wandering. It is particularly worth noticing that the author considers in Section 4 of this paper smooth maps of the annulus $A=S\sp 1\times [-a,a]$ into the larger annulus $B=S\sp 1\times [-b,b]$ which do not leave A invariant but are exact symplectic. In this setting he also obtains the existence of two fixed points. A similar but more general result was obtained by {\it Weiyue Ding} [Proc. Am. Math. Soc. 88, 341-346 (1983; Zbl 0522.55005)]. This kind of generalized Poincaré-Birkhoff theorems is very useful in applications for proving the existence of periodic solutions of some ordinary differential equations, for example, the Duffing equations.
[Ding Tongren]
MSC 2000:
*37A99 Ergodic theory
28D05 Measure-preserving transformations

Keywords: non-wandering point; twist condition; Poincaré-Birkhoff theorem; returning disks

Citations: Zbl 0522.55005

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