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We consider the homotopical dynamics on compact orientable surfaces ofpositive genus g. We establish a sufficient and necessary algebraic criterionfor homotopy classes with infinitely many periodic points of maps on suchsurfaces in terms of the characteristic polynomial of the matrix representingthe correspondig homomorphism of the first homology group. Our methods differfrom that of the literature, which typically uses Nielsen number theory,because Nielsen numbers of iterates are difficult to compute except in specialcases where they can be easily defined in terms of the easily computableLefschetz number. Instead we use some fixed point theory from the point of viewof real algebraic geometry, and some intersection theory in the context of DeRham theory on surfaces and combine it with some observations of dynamicalsystem theory which started from the work of Sarkovskii. However, in thespecial case of the homotopical dynamics of a two dimensional torus, where theNielsen number is determined by the modulus of the Lefschetz number, we recovera result in [1].

Determiniation of the homotopy class of maps on compact orientable surfaces of positive genus g with infinitely many periodic points (part I)