An index that refines the Poincaré-Lefschetz index for surface homeomorphisms
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)].
scientific article; zbMATH DE number 5712758
scientific article