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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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##### References:
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