Arnold diffusion conjecture for typical perturbations

Consider a near-integrable Hamiltonian system with slow action variables $y$ and perturbations of the form

$$H(y,x,t)=H_0(y)+\eps H_1(x,y,t)+O(\eps^2),$$

where $x\in\mathbb{T}^n$, $y\in\mathbb{R}^n$, $t\in\mathbb{T}$, and $0\leq\eps\ll 1$. Arnold diffusion conjecture. The diffusion exists for a typical set of perturbations, and the slow variables evolve along any smooth curve in the space of slow variables. Arnold diffusion concerns order-one drift of the slow variables in near-integrable systems despite small perturbations. The conjecture addresses the genericity and geometric form of this phenomenon, which are difficult to establish, especially in the real-analytic setting.