Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rüssmann’s non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus, numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang’s previous ones (1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.