Two-dimensional invariant tori in the neighborhood of an elliptic equilibrium of Hamiltonian systems

In this paper, we prove that a Hamiltonian system possesses either a four-dimensional invariant disc or an invariant Cantor set with positive (n + 2)-dimensional Lebesgue measure in the neighborhood of an elliptic equilibrium provided that its linearized system at the equilibrium satisfies some small divisor conditions. Both of the invariant sets are foliated by two-dimensional invariant tori carrying quasi-periodic solutions. Supported by NNSF of China (Grant 10531050) and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20070284004)