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A new proof of Handel’s fixed-point theorem. (Une nouvelle preuve du théorème de point fixe de Handel.) (French. English summary) Zbl 1126.37027
Summary: M. Handel has proved in [Topology 38, 235–264 (1999; Zbl 0928.55001)] 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 \(\Delta\). This index result was known to be true under an additional hypothesis and has been used by different authors (J. Franks [New York J. Math. 2, 1–19 (1996; Zbl 0891.58033) and Trans. Am. Math. Soc. 348, No. 7, 2637–2662 (1996; Zbl 0862.58006)]; S. Matsumoto [Topology Appl. 104, 191–214 (2000; Zbl 0974.37040)] to study homeomorphisms of surfaces.

MSC:
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
55M20 Fixed points and coincidences in algebraic topology
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