Orthogonal separable Hamiltonian systems on T 2

Abstract

In this paper we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T 2 for a given metric, and prove that the Hamiltonian flow on any compact level hypersurface has zero topological entropy. Furthermore, by examples we show that the integrable Hamiltonian systems on T 2 can have complicated dynamical phenomena. For instance they can have several families of invariant tori, each family is bounded by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders. As we know, it is the first concrete example to present the families of invariant tori at the same time appearing in such a complicated way.