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An equivariant foliated version of Brouwer’s translation theorem. (Une version feuilletée équivariante du théorème de translation de Brouwer.) (French. English summary) Zbl 1152.37015
L. E. J. Brouwer’s plane translation theorem [Math. Ann. 72, 37–54 (1912; JFM 43.0569.02)] tells us that for a fixed-point-free orientation-preserving homeomorphism \(f\) of the plane, every point belongs to a proper topological embedding \(C\) of \(\mathbb R\) (the so-called Brouwer lines), disjoint from its image and separating \(f(C)\) and \(f^{-1}(C)\) (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)].
The main result of the paper under review is an equivariant foliated version of Brouwer’s theorem: Let \(G\) be a discret group of orientation preserving homeomorphisms acting freely and properly on the plane. If \(f\) is a homeomorphism the Brouwer which commutes with the elements of \(G\), then there exists a \(G\)-invariant topological foliation of the plane by Brouwer lines. The previous result is applied in several ways, for instance, in the framework of area-preserving surface homeomorphisms, the author obtains a new proof of Franks’ theorem [J. M. Franks, New York J. Math. 2, 1–19, electronic (1996; Zbl 0891.58033)] which says that area-preserving two-sphere homeomorphisms having at least three fixed points always have an infinite number of periodic orbits. Another application is the following result: any Hamiltonian homeomorphism of a closed surface of genus greater or equal to \(1\) has infinitely many contractible periodic points.

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
54H20 Topological dynamics (MSC2010)
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