The asymptotic density of Wecken maps on surfaces with boundary

The Nielsen number $N(f)$ is a lower bound for the minimal number of fixed points among maps homotopic to $f$. When these numbers are equal, the map is called Wecken. A recent paper by Brimley, Griisser, Miller, and the second author investigates the abundance of Wecken maps on surfaces with boundary, and shows that the set of Wecken maps has nonzero asymptotic density. We extend the previous results as follows: When the fundamental group is free with rank $n$, we give a lower bound on the density of the Wecken maps which depends on $n$. This lower bound improves on the bounds given in the previous paper, and approaches 1 as $n$ increases. Thus the proportion of Wecken maps approaches 1 for large $n$. In this sense (for large $n$) the known examples of non-Wecken maps represent exceptional, rather than typical, behavior for maps on surfaces with boundary.