Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

Abstract

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 \[\mathcal{H}_\varepsilon(p,q,I,φ,t)=h(I)+\sum_{i=1}^n\pm \left(\frac{1}{2}p_i^2+V_i(q_i)\right)+\varepsilon H_1(p,q,I,φ, t), \] 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 $k\in [3,\infty )\cup \{\infty ,\omega \}$.

References in corpus (1)

• Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders