The generalized Lefschetz number of homeomorphisms on punctured disks

TL;DR

This paper calculates the generalized Lefschetz number for homeomorphisms on punctured disks, linking braid theory with Nielsen-Thurston classification, and identifies conditions for pseudo-Anosov behavior.

Contribution

It introduces a method to compute the Lefschetz number for punctured disk homeomorphisms using braid representations, advancing understanding of their dynamical types.

Findings

Computed Lefschetz numbers for specific braids

Determined rotation numbers for certain canonical homeomorphisms

Identified conditions under which homeomorphisms are pseudo-Anosov

Abstract

We compute the generalized Lefschetz number of orientation-preserving self-homeomorphisms of a compact punctured disk, using the fact that homotopy classes of these homeomorphisms can be identified with braids. This result is applied to study Nielsen-Thurston canonical homeomorphisms on a punctured disk. We determine, for a certain class of braids, the rotation number of the corresponding canonical homeomorphisms on the outer boundary circle. Also,it is shown that the canonical homeomorphisms corresponding to some braids are pseudo-Anosov.