The star-shaped condition on Ding’s version of the Poincaré-Birkhoff theorem. (English) Zbl 1132.54026

The famous last geometric theorem of Poincaré states that any area-preserving homeomorphism \(T\) of the annulus between two concentric circles onto itself, keeping both boundary circles invariant, and rotating them in opposite directions, has at least two fixed points. Poincaré gave a proof for some particular cases which proved to be enough for decision of the so-called billiards ball problem. The theorem was subsequently proved by Birkhoff. In order to make the theorem suitable for applications, Ding generalized it at the expense of its formulation. All of Ding’s conditions with the exception of star-shapedness were shown to be necessary. The aim of the paper under review is to prove that star-shapedness is also necessary.


54H25 Fixed-point and coincidence theorems (topological aspects)
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