An equivariant foliated version of Brouwer's translation theorem (Q820187)

\textit{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, [\textit{L. Guillou}, Topology 33, 331--351 (1994; Zbl 0924.55001)] or in [\textit{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 [\textit{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.