Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study problem following a priori unstable Hamiltonian system with time-periodic perturbation \[\mathcal{H}_\varepsilon(p,q,I,\varphi,t)=h(I)+\sum_{i=1}^n\pm \left(\frac{1}{2}p_i^2+V_i(q_i)\right)+\varepsilon H_1(p,q,I,\varphi, t), \] where $(p,q)\in \mathbb{R}^n\times\mathbb{T}^n$, $(I,\varphi)\in\mathbb{R}^d\times\mathbb{T}^d$ $n, d\geq 1$, $V_i$ are Morse potentials, and $\varepsilon$ small non-zero parameter. unperturbed not necessarily convex, induced inner dynamics does need to satisfy twist condition. Using geometric methods prove that occurs for generic perturbations $H_1$. Indeed, set admissible $H_1$ $C^\omega$ dense $C^3$ (a fortiori, open). Our perturbative technique valid $C^k$ topology all $k\in [3,\infty)\cup\{\infty, \omega\}$.