Une nouvelle preuve du théorème de point fixe de Handel
M Handel has proved in [Topology 38 (1999) 235--264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel's fixed point theorem, based on Brouwer theory and some plane topology arguments. We will slightly improve the theorem by proving the existence of a simple closed curve of index 1. This index result was known to be true under an additional hypothesis and has been used by different authors (J Franks [NYJM 2 (1996) 1--19, Trans.AMS 348 (1996) 2637--2662] S Matsumoto [Topol. Appl. 104 (2000) 191--214]) to study homeomorphisms of surfaces.
Homéomorphismes de surfaces théorèmes de la fleur de leau-fatou et de la variété stable
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Recurrent Sets for Planar Homeomorphisms
In this article, we investigate fixed-point-free orientation-preserving homeomorphisms of ℝ2 and the recurrence of connected subsets of the plane under them. In Section 1, we give a short elementary proof that an orientation-preserving homeomorphism of the plane with a periodic point must have a fixed point.