Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems

Abstract

In this paper we investigate the structure of the set of step sizes with which symplectic algorithms can simulate invariant tori of a given non-resonant frequency of general integrable Hamiltonian systems. It turns out that in the general nonlinear systems case, the set has a Cantor structure, very similar to, but more complex than, the Cantor structure of the set of Diophantine frequencies in the KAM theory. The Cantor set is of density one at the origin of the real line. Some remarks about the Cantor set and about the invariant tori are given.