  We consider the homotopical dynamics on compact orientable surfaces of
positive genus g. We establish a sufficient and necessary algebraic criterion
for homotopy classes with infinitely many periodic points of maps on such
surfaces in terms of the characteristic polynomial of the matrix representing
the correspondig homomorphism of the first homology group. Our methods differ
from that of the literature, which typically uses Nielsen number theory,
because Nielsen numbers of iterates are difficult to compute except in special
cases where they can be easily defined in terms of the easily computable
Lefschetz number. Instead we use some fixed point theory from the point of view
of real algebraic geometry, and some intersection theory in the context of De
Rham theory on surfaces and combine it with some observations of dynamical
system theory which started from the work of Sarkovskii. However, in the
special case of the homotopical dynamics of a two dimensional torus, where the
Nielsen number is determined by the modulus of the Lefschetz number, we recover
a result in [1].
