Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

Abstract

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 H ε ( p , q , I , φ , t ) = h ( I ) + ∑ i = 1 n ± 1 2 p i 2 + V i ( q i ) + ε H 1 ( p , q , I , φ , t ) , where ( p , q ) ∈ R n × T n , ( I , φ ) ∈ R d × T d with n, d ⩾ 1, V i are Morse potentials, and ɛ 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 ω dense and C 3 open (a fortiori, C ω open). Our perturbative technique for the genericity is valid in the C k topology for all k ∈ [3, ∞) ∪ {∞, ω}.