An index that refines the Poincaré-Lefschetz index for surface homeomorphisms
type: Item
instance of: scholarly article
title: An index that refines the Poincaré-Lefschetz index for surface homeomorphisms
zbMATH Open document ID: 1203.37076
author: Frédéric Le Roux
published in: Annals of Mathematics. Second Series
publication date: 2010-05-27T00:00:00Z
full work available at URL: p03.xhtml
review text: Let \(S\) be a surface, not necessarily compact, \(h:S\to S\) a homeomorphism, and \(x_0\) an isolated fixed point of \(h\). Suppose \(h\) preserves the orientation near \(x_0\), and the Poincaré-Lefschetz index of \(x_0\) is different from \(1\). The case where \(S\) is the sphere and \(x_0\) the only fixed point of \(h\) has to be excluded. In this setting, the author constructs a new invariant which is a cyclic word on the alphabet \(\{\uparrow, \rightarrow, \downarrow, \leftarrow\}\). The invariant is a refinement of the Poincaré-Lefschetz index, and it is invariant under conjugation, but it is not an invariant of the local dynamics of \(h\) near \(x_0\). It reflects a certain canonical decomposition of the dynamics into a finite number of sectors of hyperbolic, elliptic, or indifferent type. The proof is based on ideas behind Brouwer's plane translation theorem, as presented by the author in [Astérisque 292. Paris: Société Mathématique de France (2004; Zbl 1073.37046)].
reviewed by: Thomas J. Bartsch
Mathematics Subject Classification ID: 37E30
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37C25
zbMATH DE Number: 5712758
zbMATH Keywords: surface homeomorphism
zbMATH Keywords: fixed point
zbMATH Keywords: Poincaré-Lefschetz index
zbMATH Keywords: topological dynamics
MaRDI profile type: Publication
OpenAlex ID: W2370264416
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DOI: 10.4007/ANNALS.2010.171.1531
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description: scientific article; zbMATH DE number 5712758