Generalizations of the Poincaré-Birkhoff theorem. (English) Zbl 0676.58037

Ann. Math. (2) 128, No. 1, 139-151 (1988); erratum 164, 1097-1098 (2006).
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 \(A=S^ 1\times [-a,a]\) into the larger annulus \(B=S^ 1\times [-b,b]\) 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 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.
Reviewer: Ding Tongren


37A99 Ergodic theory
28D05 Measure-preserving transformations


Zbl 0522.55005
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