ON PERIODIC POINTS OF SYMPLECTOMORPHISMS ON SURFACES

We construct a symplectic flow on a surface of genus g >= 2, Sigma(g >= 2), with exactly 2g - 2 hyperbolic fixed points and no other periodic orbits. Moreover, we prove that a (strongly nondegenerate) symplectomorphism of Sigma(g >= 2) isotopic to the identity has infinitely many periodic points if there exists a fixed point with nonzero mean index. From this result, we obtain two corollaries, namely that such a symplectomorphism of Sigma(g >= 2) with an elliptic fixed point or with strictly more than 2g - 2 fixed points has infinitely many periodic points provided that the flux of the isotopy is "irrational".