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 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}.

Arnold diffusion in Hamiltonian systems on infinite lattices

“…Remark It is well known that the Hypotheses $$\mathbf {H3}$$ and $$\mathbf {H4}$$ are $$C^r$$ (with $$r\ge 4$$) and $$C^\infty$$ generic. They are also generic in the analytic setting if one considers sufficiently small $$h>0$$ (see [16]). On the contrary the Hypothesis $$\mathbf {H2}$$ requires that certain subspaces are invariant and the dynamics on them is integrable.…”

Arnold diffusion in Hamiltonian systems on infinite lattices

“…It is well known that they are C r (with r ≥ 4) and C ∞ generic. They are also generic in the analytic setting if one considers sufficiently small h > 0 (see [14]). On the contrary the Hypothesis H2 requires that certain subspaces are invariant and the dynamics on them is integrable.…”