Symplectic Diffeomorphisms with Infinitely Many Elliptic Periodic Points

Let M be a compact manifold of dimension greater than one. Let Λ fM be a non trivial basic set for f∈ Diff r(M) (r≥2). Suppose p is a hyperbolic periodic point of f in Λ f with period k, det T pf k<1, dim W u(p,f)=1, W s(p,f) and W u(p,f) have a point of tangency. Newhouse[1] shows that there exists a residual subset B of a neighborhood N of f such that for each g∈ B, g has infinitely many sinks. Later, on basis of the conclusion of Takens, Newhouse also proved that for a compact symplectic manifold M, there exists a residual subset BDiff 1 ω(M) such that if f∈ B, then either f is Anosov or 1 elliptic periodic points of f are dense on M. A periodic point p is called a 1 elliptic periodic point if two eigenvalues of df k p(k is period of p) have norm 1 and the others are not. However, whether there exists elliptic periodic points or not is a more interesting problem, and near the elliptic periodic point the dynamical behavior of the symplectic diffeomorphism is more complicate also. If the dimension of M is 2, the conclusion of Newhouse implies the density of the elliptic periodic points of f. The same problem remains open in higher dimensional case. In this paper, we show that there exists a class of higher dimensional symplectic diffeomorphisms which have infinitely many elliptic periodic points.