The generalized Lefschetz number of homeomorphisms on punctured disks. (English) Zbl 1189.37050

For orientation-preserving self-homeomorphisms \(f\) of a compact punctured disk which preserve the outer boundary circle, the generalized Lefschetz number \(\mathcal L(f)\) is computed using the fact that homotopy classes of these homeomorphisms can be identified with braids. Such homeomorphisms are important in the topological study of 2-dimensional dynamical, they include the homeomorphisms which are obtained from orientation-preserving disk homeomorphisms by the blow-up construction at a finite, interior invariant set. The result obtained is applied to study Nielsen-Thurston canonical homeomorphisms on a punctured disk. In particular, for a certain class of braids, the rotation number of the corresponding canonical homeomorphisms on the outer boundary circle is determined. As a consequence of the result on the rotation number, it is shown that the canonical homeomorphisms corresponding to some braids are pseudo-Anosov with associated foliations having no interior singularities. A lower and an upper bound for the Nielsen number \(N(f)\) for a certain class of braids is given.


37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
55M20 Fixed points and coincidences in algebraic topology
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