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Handel’s fixed point theorem: a Morse theoretical point of view. (English) Zbl 1513.37030

Summary: M. Handel has proved in [Topology 38, No. 2, 235–264 (1999; Zbl 0928.55001)] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that turned out to be an efficient tool in the study of the dynamics of surface homeomorphisms. The present article fits into a series of articles by the author [Geom. Topol. 10, 2299–2349 (2006; Zbl 1126.37027)] and by J. Xavier [Fundam. Math. 219, No. 1, 59–96 (2012; Zbl 1277.37028); Ergodic Theory Dyn. Syst. 33, No. 5, 1584–1610 (2013; Zbl 1288.54047)], where proofs were given, related to the classical Brouwer Theory, instead of the Homotopical Brouwer Theory used in the original article. Like in [the author, loc. cit; Xavier, loc. cit. (2012); loc. cit. (2013)], we will use “free brick decompositions” but will present a more conceptual Morse theoretical argument. It is based on a new preliminary lemma, that gives a nice “condition at infinity” for our problem.

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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