具有渐近稠密投影轨道的$\mathbb{A}^3$上几乎可积系统的例子
Examples of nearly integrable systems on $\mathbb{A}^3$ with asymptotically dense projected orbits
摘要：
给定一个整数$κ\geq2$，我们在$κ\geq2$上引入一类几乎可积系统，其形式为$$ H_n(θ,r)=\frac12 \Vert r\Vert ^2+\tfrac{1}{n} U(θ_2,θ_3)+f_n(θ,r) $$，其中$U\in C^κ(\mathbb{T}^2)$是一般的势函数，$f_n$是$C^{κ-1}$的附加扰动，使得$\Vert f_n\Vert_{C^{κ-1}(\mathbb{A}^3)}\leq \tfrac{1}{n}$，从而$H_n$是完全可积系统$h(r)=\frac12\Vert r\Vert ^2$的扰动.设$Π:\mathbb{A}^3\to\mathbb{R}^3$为正则投影。我们证明，对于每个$δ>0$，存在$n_0$，使得对于$n\geq n_0$，系统$H_n$承认能量为$\frac12$的轨道$Γ_n$，其投影$Π(Γ_n)$是$δ$ -在$Π(H_n^{-1}(\tfrac{1}{2}))$中稠密的，从这个意义上说，$δ$-$\mathbb{R}^3$中$Π(Γ_n)$的邻居覆盖了$Π(H_n^{-1}(\frac{1}{2}))$。
英文摘要：
Given an integer $κ\geq2$, we introduce a class of nearly integrable systems on $\mathbb{A}^3$, of the form $$ H_n(θ,r)=\frac12 \Vert r\Vert ^2+\tfrac{1}{n} U(θ_2,θ_3)+f_n(θ,r) $$ where $U\in C^κ(\mathbb{T}^2)$ is a generic potential function and $f_n$ a $C^{κ-1}$ additional perturbation such that $\Vert f_n\Vert_{C^{κ-1}(\mathbb{A}^3)}\leq \tfrac{1}{n}$, so that $H_n$ is a perturbation of the completely integrable system $h(r)=\frac12\Vert r\Vert ^2$. Let $Π:\mathbb{A}^3\to\mathbb{R}^3$ be the canonical projection. We prove that for each $δ>0$, there exists $n_0$ such that for $n\geq n_0$, the system $H_n$ admits an orbit $Γ_n$ at energy $\frac12$ whose projection $Π(Γ_n)$ is $δ$-dense in $Π(H_n^{-1}(\tfrac{1}{2}))$, in the sense that the $δ$-neighborhood of $Π(Γ_n)$ in $\mathbb{R}^3$ covers $Π(H_n^{-1}(\frac{1}{2}))$.
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