The genericity, freedom and velocity conjectures for Arnold diffusion

Let $H=H_0+\varepsilon H_1+\varepsilon^2H_2$ be the nonautonomous near-integrable Hamiltonian system described above, with normally hyperbolic invariant manifold $N$ foliated by invariant tori $N_y$ and separatrices forming a figure-eight. A diffusion trajectory is a trajectory whose projection to the $y$-space makes a substantial displacement through the domain $\mathcal D$.

Arnold diffusion conjecture. For sufficiently large $r\in{\mathbb N}\cup\\{\infty,\omega\\}$ and an open dense set of $C^r$ perturbations: (A) diffusion exists; (B) the projection of a diffusion trajectory to the $y$-space can remain in a small neighborhood of any smooth curve $\chi\subset\mathcal D$; and (C) there are fast diffusion trajectories whose average velocity along $\chi$ is of order $\varepsilon/\log\varepsilon$.

These are the genericity, freedom and velocity aspects of Arnold diffusion in the a priori unstable case. The statement concerns the existence and controllability of diffusion trajectories near the asymptotic manifolds of the normally hyperbolic invariant manifold, together with the conjectured fastest scale of diffusion.