Nielsen periodic point theory for periodic maps on orientable surfaces
The authors are engaged in a long-term project to study the periodic point behavior of self-maps of manifolds, under homotopies of the map, by means of Nielsen periodic point theory. In a previous paper [Topology Appl. 95, 1--30 (1999; Zbl 1007.55002)], they demonstrated that, even for maps of the orientable hyperbolic surface of lowest genus, the calculation of the Nielsen periodic point numbers can be a formidable challenge. In the present paper, the setting is that of all orientable hyperbolic surfaces but, instead of arbitrary maps, they consider periodic homeomorphisms. Let \(f\) be periodic of period \(m\) on the surface of genus \(g > 1\). If the Lefschetz numbers of the \(k\)-th iterates of \(f\) are zero for all \(k < m\), then \(f\) is said to be minLef. It is proved that a necessary condition for \(f\) to be minLef is that its period \(m\) must be even and it must divide the Euler characteristic of the surface, that is, \(2 - 2g\). A minLef homeomorphism \(f\) may have periodic points of a period \(k < m\) where \(k\) divides \(m\) and then \(f\) is said to be reducible minLef. Otherwise, it is irreducible. The main results of the paper calculate the Nielsen periodic point numbers for each of the three cases: reducible minLef, irreducible minLef and non-minLef. Well-chosen examples illustrate the periodic point behavior in each case. A key to the calculation of the Nielsen periodic point numbers in this setting is the authors' discovery that the geometric property of reducibility of \(f\) is captured by the fundamental group isomorphism induced by \(f\). Moreover, the detailed analysis of fixed points of homeomorphisms of hyperbolic surfaces by \textit{B. Jiang} and \textit{J. Guo} [Pac. J. Math. 160, 67--89 (1993; Zbl 0829.55001)] furnishes crucial geometric information about the structure of the set of periodic points of \(f\) of each period. This impressive and very well-written paper uses sophisticated tools to study the periodic point theory of an important class of homeomorphisms of surfaces. Since the periodic point theory of pseudo-Anosov homeomorphisms is well understood and much simpler than is the case for periodic homeomorphisms, the paper is a significant step toward the goal of understanding the periodic point theory of all homeomorphisms of hyperbolic surfaces.
scientific article; zbMATH DE number 5017442