标题： 在完全可积系统中生成双曲奇点 显示英文标题

标题： Generating hyperbolic singularities in completely integrable systems

摘要： 设$(M,\Omega)$是一个连通的辛 4-流形，令$F=(J,H) : M \to \mathbb{R}^2$是在$M$上的一个完全可积系统，其仅有非退化奇点，并且$J : M \to \mathbb{R}$是一个正则映射。 假设$F$不包含具有双曲块的奇点，并且$p_1,...,p_n$是$F$的焦点-焦点奇点。 对于每个子集$S=\{i_1,...,i_j\}$，我们将展示如何在任何$p_i, i \in S$附近局部修改$F$，以创建一个新的可积系统$\tilde{F}=(J, \tilde{H}) : M \to \mathbb{R}^2$，使得其经典谱$\tilde{F}(M)$包含$j$条对应于$\tilde{F}$的非退化横向双曲奇点的光滑奇异值曲线。 此外，$\tilde{F}$的焦点-焦点奇点恰好是$p_i$、$i \in \{1,...,n\} \setminus S$，且每个$p_i$都是非退化的。 证明基于非退化奇点的Eliasson线性化定理以及哈密顿霍普夫分岔的性质。 显示英文摘要

摘要： Let $(M,\Omega)$ be a connected symplectic 4-manifold and let $F=(J,H) : M \to \mathbb{R}^2$ be a completely integrable system on $M$ with only non-degenerate singularities and for which $J : M \to \mathbb{R}$ is a proper map. Assume that $F$ does not have singularities with hyperbolic blocks and that $p_1,...,p_n$ are the focus-focus singularities of $F$. For each subset $S=\{i_1,...,i_j\}$ we will show how to modify $F$ locally around any $p_i, i \in S$, in order to create a new integrable system $\tilde{F}=(J, \tilde{H}) : M \to \mathbb{R}^2$ such that its classical spectrum $\tilde{F}(M)$ contains $j$ smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of $\tilde{F}$. Moreover the focus-focus singularities of $\tilde{F}$ are precisely $p_i$, $i \in \{1,...,n\} \setminus S$, and each of these $p_i$ is non-degenerate. The proof is based on Eliasson's linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.