Homoclinic points in symplectic and volume-preserving diffeomorphisms

LetMⁿ be a compactn-dimensional manifold and ω be a symplectic or volume form onMⁿ. Let ϕ be aC¹ diffeomorphism onMⁿ that preserves ω and letp be a hyperbolic periodic point of Φ. We show that genericallyp has a homoclinic point, and moreover, the homoclinic points ofp is dense on both stable manifold and unstable manifold ofp. Takens [11] obtained the same result forn=2.