Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems
Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems
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DOI：
10.1088/1361-6544/ac50bb
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发表时间：
2021-03
期刊：
影响因子：
1.7
通讯作者：
Qinbo Chen;R. de la Llave
中科院分区：
文献类型：
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作者：
Qinbo Chen;R. de la Llave
Arnold扩散在解析范畴中的普适性是一个公开问题。本文研究了具有时间周期扰动的先验不稳定Hamilton系统Hε（p，q，I，φ，t）=h（I）+∑i= 1 n ± 12 pi 2 +Vi（qi）+εH1（p，q，I，φ，t）中的非线性问题，其中（p，q）∈Rn×Tn，（I，φ）∈Rd×Td，n，d莫尔斯势，Vi是一个非零小参数.未扰动的哈密顿量不一定是凸的，并且诱导的内部动力学不需要满足扭曲条件。使用几何方法，我们证明了阿诺德扩散发生一般的解析扰动H1。事实上，容许H1的集合是C ω稠密和C3开的（更不用说C ω开了）。我们的一般性微扰技巧在Ck拓扑中对所有k ∈ [3，∞）<${∞，ω}都是有效的.
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 Hε(p,q,I,φ,t)=h(I)+∑i=1n±12pi2+Vi(qi)+εH1(p,q,I,φ,t), where (p,q)∈Rn×Tn , (I,φ)∈Rd×Td 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}.