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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
##### Keywords:
homeomorphisms; two-torus; Conley-Zehnder theorem
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