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Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

https://doi.org/10.1088/1361-6544/ac50bb

Chen, Qinbo; de la Llave, Rafael (March 2022, Nonlinearity)

Abstract The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the followinga prioriunstable Hamiltonian system with a time-periodic perturbation

H_{ε} (p, q, I, φ, t) = h (I) + \sum_{i = 1}^{n} \pm (\frac{1}{2} p_{i}^{2} + V_{i} (q_{i})) + ε H_{1} (p, q, I, φ, t),

where

(p, q) \in R^{n} \times T^{n}

(I, φ) \in R^{d} \times T^{d}

withn,d⩾ 1,V_iare Morse potentials, andɛis 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 perturbationsH₁. Indeed, the set of admissibleH₁isC^ωdense andC³open (a fortiori,C^ωopen). Our perturbative technique for the genericity is valid in theC^ktopology for allk∈ [3, ∞) ∪ {∞,ω}.

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