  We use Wagner's algorithm to estimate the number of periodic points of
certain selfmaps on compact surfaces with boundary. When counting according to
homotopy classes, we can use the asymptotic density to measure the size of sets
of selfmaps. In this sense, we show that "almost all" such selfmaps have
periodic points of every period, and that in fact the number of periodic points
of period n grows exponentially in n. We further discuss this exponential
growth rate and the topological and fundamental-group entropies of these maps.
  Since our approach is via the Nielsen number, which is homotopy and
homotopy-type invariant, our results hold for selfmaps of any space which has
the homotopy type of a compact surface with boundary.
