Arnold's diffusion conjecture for nearly integrable Hamiltonian systems

Let $h$ be an integrable Hamiltonian on the action space, let $P$ be a generic perturbation, and consider the nearly integrable Hamiltonian

$$H(x,y)=h(y)+\varepsilon P(x,y),\qquad (x,y)\in\mathbb T^n\times D.$$

For any two points $y'$ and $y”$ on the connected level hypersurface of $h$ in the action space, Arnold's diffusion conjecture. there exist orbits of $H$ connecting an arbitrarily small neighborhood of the torus $y=y'$ with an arbitrarily small neighborhood of the torus $y=y”$, provided that $\varepsilon\neq 0$ is sufficiently small and $P$ is generic.

This conjecture predicts large-scale drift in the action variables of nearly integrable systems with at least three degrees of freedom, beyond the confinement supplied by invariant KAM tori. The source presents it as Arnold's conjecture; its resolution status is not specified here.