Almost dense orbit on energy surface

The famous question called the ergodic hypothesis suggested that for a typical Hamiltonian on a typical energy surface all, but a set of zero measure of initial conditions, have trajectories covering densely this energy surface itself. However, KAM theory showed that for nearly integrable systems there is a set of initial conditions of positive measure of quasi periodic trajectories. This disproved the ergodic hypothesis and forced to reconsider the problem. A quasi ergodic hypothesis asks if a typical Hamiltonian on a typical energy surface has a dense orbit. A definite answer whether this statement is true or not is still far out of reach of modern dynamics. There was an attempt to prove this statement by E. Fermi,5 which failed (see6 for more detailed account). To simplify the quasi ergodic hypothesis, M. Herman7 formulated the following question: Can one find an example of a C∞ Hamiltonian H in a C small neighborhood of H0(p) = 〈p,p〉 2 such that on the unit energy surface {H −1( 2 )} there is a dense trajectory? Many people believe that such examples do exist and are C∞–generic (see,4,31). In this paper we make a step in the direction of answering Herman’s question. For any r we construct a Hamiltonian, which is C close to H0(p) = 〈p,p〉 2 and has a trajectory dense in a set of Lebesgue measure 1/2 on the energy surface. Here is the exact statement. Let q ∈ T3, p ∈ R3 and H0(p) = 〈p,p〉 2 be the unperturbed Hamiltonian, where 〈p, p〉 is the dot product in R3. Theorem 1.1. For any r ≥ 2 there is a C small perturbation Hε(q, p) = H0(p) + !H1(q, p, !) and an orbit (q(t), p(t)) : R → T3 × R3 of