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The purpose of this mostly expository paper is to discuss a connection   between Nielsen fixed point theory and symplectic Floerhomology for symplectomorphisms of surfaces and a calculation  of  Seidel’s symplectic Floer homology for different mappingclasses. We also describe symplectic zeta functions and an  asymptotic  symplectic invariant. A generalisation of the  Poincaré-Birkhofffixed point theorem and Arnold conjecture is proposed

Nielsen theory, Floer homology and a generalisation of the Poincaré-Birkhoff theorem