AbstractWe prove a form of Arnold diffusion in the a-priori stable case. Let $$H_{0}(p)+\epsilon H_{1}(\theta,p,t),\quad \theta \in {\mathbb{T}^{n}},\,p \in B^{n},\,t \in \mathbb{T}= \mathbb{R}/\mathbb{T},$$H0(p)+ϵH1(θ,p,t),θ∈Tn,p∈Bn,t∈T=R/T,be a nearly integrable system of arbitrary degrees of freedom $${n \geqslant 2}$$n⩾2 with a strictly convex H0. We show that for a “generic” $${\epsilon H_1}$$ϵH1, there exists an orbit $${(\theta,p)}$$(θ,p) satisfying $$\|p(t)-p(0)\| > l(H_{1}) > 0,$$‖p(t)-p(0)‖>l(H1)>0,where $${l(H_1)}$$l(H1) is independent of $${\epsilon}$$ϵ. The diffusion orbit travels along a codimension-1 resonance, and the only obstruction to our construction is a finite set of additional resonances.For the proof we use a combination of geometric and variational methods, and manage to adapt tools which have recently been developed in the a-priori unstable case.

中文翻译：

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任意自由度和正常双曲不变圆柱的阿诺德扩散

摘要 我们在先验稳定的情况下证明了 Arnold 扩散的一种形式。让 $$H_{0}(p)+\epsilon H_{1}(\theta,p,t),\quad \theta \in {\mathbb{T}^{n}},\,p \in B ^{n},\,t \in \mathbb{T}= \mathbb{R}/\mathbb{T},$$H0(p)+ϵH1(θ,p,t),θ∈Tn,p∈ Bn,t∈T=R/T，是一个具有严格凸 H0 的任意自由度 $${n \geqslant 2}$$n⩾2 的近似可积系统。我们证明对于“通用”$${\epsilon H_1}$$ϵH1，存在满足 $$\|p(t)-的轨道 $${(\theta,p)}$$(θ,p) p(0)\| > l(H_{1}) > 0,$$‖p(t)-p(0)‖>l(H1)>0，其中 $${l(H_1)}$$l(H1) 独立于$${\epsilon}$$ϵ。扩散轨道沿着 codimension-1 共振传播，我们构建的唯一障碍是一组有限的额外共振。 为了证明，我们使用几何和变分方法的组合，