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A generalization of the Conley-Zehnder theorem to homeomorphisms of the two-dimensional torus. (Une généralisation du théorème de Conley-Zehnder aux homéomorphismes du tore de dimension deux.) (French) Zbl 0877.58011
A result due to C. C. Conley and E. Zehnder [Invent. Math. 73, 33-49 (1983; Zbl 0516.58017)] implies that under some natural conditions a measure preserving diffeomorphism on the two-torus has three different fixed points at least. The present paper extends this result to homeomorphisms.
The following theorem is proved: If \(F\) is a homeomorphism of the two-torus which is homotopic to the identity and preserves the Lebesgue measure, and if \(f\) is a lift of \(F\) to \(\mathbb{R}^2\) which preserves the centre of mass, then \(f\) has three distinct fixed points, at least, on the torus.
The proof is difficult and differs completely from the proof of different variants of the Conley-Zehnder theorem.

MSC:
58C30 Fixed-point theorems on manifolds
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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