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A dynamical proof of Brouwer’s translation theorem. (Une démonstration dynamique du théorème de translation de Brouwer.) (French) Zbl 0859.54029
The Brouwer translation theorem asserts that if \(f\) is a fixed point free orientation preserving homeomorphism of the plane, then every point belongs to a curve \({\mathcal C}\), image of a proper topological embedding of \(\mathbb{R}\), such that \({\mathcal C}\) does not meet \(f({\mathcal C})\) and separates \(f({\mathcal C})\) and \(f^{-1}({\mathcal C})\). The Brouwer translation theorem has many applications in two-dimensional topological dynamics. An example is the geometrical theorem of Poincaré-Birkhoff, which asserts that a homeomorphism \(F\) of a ring, preserving area, orientation and two boundaries of the ring, and satisfying the property of “twist” on the boundaries, has at least two fixed points. The authors use ideas of topological dynamics to prove the theorem. They are proving the theorem from the fact, that the homeomorphism \(f\) has no periodic orbit, and they use a property of free disk chains and a decomposition of the plane in free subsets.

54H20 Topological dynamics (MSC2010)
37B99 Topological dynamics