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Structure of Brouwer homeomorphismus. (Structure des homéomorphismes de Brouwer.) (French. English summary) Zbl 1087.37035

A Brouwer homeomorphism is a homeomorphism of the plane that preserves the orientation and with no fixed-points. Given a Brouwer homeomorphism \(h\), a subset \(U\) of the plane is said to be a translation domain (for \(h\)) if \(U\) is an open, connected and simply connected set which is invariant for \(h\) and such that the restriction of \(h\) to \(U\) is conjugated to the translation \(\tau \) of the plane defined by the requirement \(\tau (x,y)\) be \((x+1,y)\). A result of Brouwer states that every point of the plane belongs to a translation domain.
The author presents the following result: for every Brouwer homomorphism \(h\), there exists a covering of the plane by translation domains. Given a Brouwer homomorphism \(h\), the author defines a distance \(d_{h}\), on the plane linked to \(h\), named \(h\)-distance or translation distance of \(h\), in the following way: if \(x\) and \(y\) are two points of the plane, \(d_{h}(x,y)\) is the smallest integer \(k\) such that there are translation domains \(O_{1}, \dots ,O_{k}\) such that \(\bigcup_{n=1}^{k}O_{n}\) is a connected set containing both \(x\) and \(y\). By means of the distance \(d_{h}\) a combinatorial conjugacy is presented and the following result is proved: if \(d_{h}(x,y)>1\), there is a finite family of generalized Reeb components separating \(x\) and \(y\).

MSC:

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37B30 Index theory for dynamical systems, Morse-Conley indices
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References:

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