  The purpose of this mostly expository paper is to discuss a connection
between Nielsen fixed point theory and symplectic Floer homology theory for
symplectomorphisms of surface and a calculation of Seidel's symplectic Floer
homology for different mapping classes. We also describe symplectic zeta
functions and asympltotic symplectic invariant. A generalisation of the
Poincare- Birkhoff fixed point theorem and Arnold conjecture is proposed.
