The generic symplectic C^{1}-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point

Abstract

We prove that if (M,\omega) is a connected and compact four-dimensional symplectic manifold, there exist three open sets U_1, U_2, U_3 of {\rm Diff}^1_{\omega}(M) (for the C^1 topology) such that: U_1\cup U_2\cup U_3 is dense in {\rm Diff}^1_{\omega}(M);f\in U_1 if and only if f is Anosov and transitive;f\in U_2 if and only if f is partially hyperbolic; andf\in U_3 if and only if f has a stable completely elliptic periodic point.