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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.

MSC:
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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