  We show that S. Saito's fixed point formula serves as a powerful tool for
counting the number of isolated periodic points of an area-preserving surface
map admitting periodic curves. His notion of periodic curves of types I and II
plays a central role in our discussion. We establish a Shub-Sullivan type
result on the stability of local indices under iterations of the map, the
finiteness of the number of periodic curves of type II, and the absence of
periodic curves of type I. Combined with these results, Saito's formula implies
the existence of infinitely many isolated periodic points whose cardinality
grows exponentially as period tends to infinity.
