论文标题
在某些可逆系统中不变的托里
On invariant tori in some reversible systems
论文作者
论文摘要
在本文中，我们考虑以下可逆系统\ begin {equation*} \ begin {cases} \ dot {x} =ω_0+f（x，x，y），\\ \ dot {y} = g（x，x，y），\ end end end eNd {cases}} $ y \ backsim0 \ in \ mathbf {r}^{d} $，$ω_0$是二芬太丁，$ f（x，y）= o（y）$，$ g（x，x，y）= o（y^2）$和$ f $，$ f $，$ g $相对于IDIFUTION g：$ $ $（x，y），x，y），y（x，y） $ f（-x，y）= f（x，y）$，$ g（-x，y）= - g（x，y）$。我们研究了其他不变的托里（Tori）的分析不变圆环$γ_0$γ_0$ $γ_0$的积累。我们将证明，如果Birkhoff的正常形式左右$γ_0$是0级的，那么$γ_0$是由其他分析不变的Tori累积的，那么这些Tori的结合的Lebesgue度量为正，并且这些tori tori的密度为$γ_0$。我们还将证明，如果Birkhoff的正常形式约为$γ_0$是$ j $ -Degenate（$ 1 \ leq j \ leq d-1 $）和条件（1.6），那么通过$γ_0$通过$γ_0$通过dimention $ d $ d+j $ d $ d d+j $ d+j $ d+j $ d+j $ fofector $ j $ foli fore频率vector $ $ $ω_0。如果birkhoff的普通形式左右$γ_0$是$ d-1 $ - 定型，我们将证明一个更强的结果，也就是说，将$γ_0$的完整社区散布在分析不变的Tori中，其频率向量与$ω_0$成比例。
In the present paper, we consider the following reversible system \begin{equation*} \begin{cases} \dot{x}=ω_0+f(x,y),\\ \dot{y}=g(x,y), \end{cases} \end{equation*} where $x\in\mathbf{T}^{d}$, $y\backsim0\in \mathbf{R}^{d}$, $ω_0$ is Diophantine, $f(x,y)=O(y)$, $g(x,y)=O(y^2)$ and $f$, $g$ are reversible with respect to the involution G: $(x,y)\mapsto(-x,y)$, that is, $f(-x,y)=f(x,y)$, $g(-x,y)=-g(x,y)$. We study the accumulation of an analytic invariant torus $Γ_0$ of the reversible system with Diophantine frequency $ω_0$ by other invariant tori. We will prove that if the Birkhoff normal form around $Γ_0$ is 0-degenerate, then $Γ_0$ is accumulated by other analytic invariant tori, the Lebesgue measure of the union of these tori being positive and the density of the union of these tori at $Γ_0$ being one. We will also prove that if the Birkhoff normal form around $Γ_0$ is $j$-degenerate ($1\leq j\leq d-1$) and condition (1.6) is satisfied, then through $Γ_0$ there passes an analytic subvariety of dimension $d+j$ foliated into analytic invariant tori with frequency vector $ω_0$. If the Birkhoff normal form around $Γ_0$ is $d-1$-degenerate, we will prove a stronger result, that is, a full neighborhood of $Γ_0$ is foliated into analytic invariant tori with frequency vectors proportional to $ω_0$.
