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Free lines for homeomorphisms of the open annulus. (English) Zbl 1135.37016
In this paper the properties of the fixed point free orientation preserving homeomorphisms of the plane (so-called Brouwer homeomorphisms) are studied. Further, a Brouwer line \(L\) for the Brouwer homeomorphisms \(h\) is the image of a proper embedding of \(\mathbb{R}\) into \(\mathbb{R}^2\) such that \(L\) is free under \(h\), (i.e. \(h(L)\bigcap L =\emptyset\)) and \(L\) separates \(h(L)\) and \(h^{-1}(L)\). The author proves the following result: Let \(H\) be a homeomorphism of the open annulus \(S^1 \times\mathbb{R}\) isotopic to the identity which admits a lift \(h\) to \(\mathbb{R}^2\) without fixed point. Then there exists an essential simple closed curve in the annulus free under \(H\) ( and therefore it lifts to a Brouwer line for \(h\)) or there exists a properly imbedded line in the annulus joining both ends which lifts to a Brouwer line for \(h\). Also, the connection with the Poincaré-Birkhoff theorem is discussed.

MSC:
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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