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A foliated version of the Brouwer translation theorem. (Une version feuilletée du théorème de translation de Brouwer.) (French. English summary) Zbl 1073.37045
A celebrated theorem due to Brouwer states that for a fixed-point-free orientation-preserving homeomorphism \(f\) of the plane, every point belongs to a proper topological imbedding \(C\) of \(\mathbb{R}\) (the so-called Brouwer lines), disjoint from its image and separating \(f(C)\) and \(f^{-1}(C)\) [L. E. J. Brouwer, Math. Ann. 72, 37–54 (1912; JFM 43.0569.04)] (more recent proofs of Brouwer’s theorem are available in, for instance, [L. Guillou, Topology 33, 331–351 (1994; Zbl 0924.55001)] or in [J. Franks, Ergodic Theory Dyn. Syst. 12, 217–226 (1992; Zbl 0767.58025)]. Brouwer’s theorem has been applied in several fields of mathematics, for example in dynamical systems. For several types of Brouwer homeomorphisms (i.e., homeomorphisms satisfying the hypothesis of Brouwer’s theorem), it is known that one can construct a foliation of the plane by means of its Brouwer lines.
The main result of the paper under review states that this situation is general: Theorem. If \(f\) is a Brouwer homeomorphism, then a foliation of the plan can be built by using the Brouwer lines associated to \(f\). The main tool used in the proof of the foregoing theorem is a decomposition of the plane in free and maximal subsets following an idea of M. Flucher [Manuscr. Math. 68, 271–293 (1990; Zbl 0722.58027)] developed by A. Sauzet in [Application des décompositions libres à l’étude des homéomorphismes de surface, Thése de l’Université Paris 13 (2001)].

MSC:
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37E35 Flows on surfaces
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