论文标题
Kolmogorov的归化汉密尔顿系统的定理具有连续参数
Kolmogorov's Theorem for Degenerate Hamiltonian Systems with Continuous Parameters
论文作者
论文摘要
在本文中，我们研究了Kolmogorov型定理，用于堕落的汉密尔顿系统的小扰动。这些系统由参数$ξ$ as \（y，x，x，ξ）= \langleΩ（ξ），y \ rangle + \ varepsilon p（y，x，x，x，\ varepsilon）\）索引。我们假设频率映射（$ω$）相对于$ξ$是连续的。此外，扰动函数$ p（y，x，cdot，\ varepsilon）$，保持Hölder的连续性约为$ξ$。我们证明，在某些拓扑度条件下，持续不变的Tori保留了与未扰动的Tori的频率相同的频率，并且用于频率映射的凸状条件弱。值得注意的是，在我们的理解中，本文提出了开创性的kam定理，在这种情况下仅假设频率映射$ω$对参数的连续依赖性。
In this paper, we investigate Kolmogorov type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter $ξ$ as \( H(y,x,ξ) = \langleω(ξ),y\rangle + \varepsilon P(y,x,ξ,\varepsilon) \) where $\varepsilon>0$. We assume that the frequency map, $ω$, is continuous with respect to $ξ$. Additionally, the perturbation function, $P(y,x,\cdot, \varepsilon)$, maintains Hölder continuity about $ξ$. We prove that persistent invariant tori retain the same frequency as those of the unperturbed tori, under certain topological degree conditions and a weak convexity condition for the frequency mapping. Notably, this paper presents, to our understanding, pioneering results on the KAM theorem under such conditions-with only assumption of continuous dependence of frequency mapping $ω$ on the parameter.
