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