## 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.

### MSC:

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

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