arXiv 0909.38012009-09-21math.DS

关于哈密顿量的双曲闭轨集的稳定性

On the Stability of the Set of Hyperbolic Closed Orbits of a Hamiltonian

Mario Bessa, Celia Ferreira, Jorge Rocha

DOI：: 10.1017/S0305004110000253

摘要：

一个哈密尔顿能级，比如一个哈密尔顿能级$H$和一个能量$e \in \mathbb{R}$的一对$(H,e)$，被称为Anosov，如果存在$H^{-1}({e})$的连通分量$\mathcal{E}_{H,e}$，它对于哈密尔顿流$X_H^t$是一致双曲的。如果存在能级$H^{-1}({e})$的连通分量$\mathcal{E}^\star_{H,e}$，使得$\mathcal{E}^\star_{H,e}$的所有闭合轨道和所有临界点都是双曲的，则称$(H,e)$对是哈密尔顿星系统，并且对于任何哈密尔顿算子$\tilde{H}$，在$H$的某个$C^2$ -邻域中，对于能级$\tilde{H}^{-1}({\tilde{e}})$的连通分量，接近于$\mathcal{E}^\star_{H,e}$，和$\tilde{e}$在$e$的某个邻域中。本文证明了：对于任意四维Hamilton星能级$(H,e)$，若表面$\mathcal{E}^\star_{H,e}$不含临界点，则$X_H^t|_{\mathcal{E}^\star_{H,e}}$是Anosov;若表面$\mathcal{E}^\star_{H,e}$有临界点，则存在任意接近于$e$的$\tilde{e}$，使得$X_H^t|_{\mathcal{E}^\star_{H,\tilde{e}}}$是Anosov.

英文摘要：

A Hamiltonian level, say a pair $(H,e)$ of a Hamiltonian $H$ and an energy $e \in \mathbb{R}$, is said to be Anosov if there exists a connected component $\mathcal{E}_{H,e}$ of $H^{-1}({e})$ which is uniformly hyperbolic for the Hamiltonian flow $X_H^t$. The pair $(H,e)$ is said to be a Hamiltonian star system if there exists a connected component $\mathcal{E}^\star_{H,e}$ of the energy level $H^{-1}({e})$ such that all the closed orbits and all the critical points of $\mathcal{E}^\star_{H,e}$ are hyperbolic, and the same holds for a connected component of the energy level $\tilde{H}^{-1}({\tilde{e}})$, close to $\mathcal{E}^\star_{H,e}$, for any Hamiltonian $\tilde{H}$, in some $C^2$-neighbourhood of $H$, and $\tilde{e}$ in some neighbourhood of $e$. In this article we prove that for any four-dimensional Hamiltonian star level $(H,e)$ if the surface $\mathcal{E}^\star_{H,e}$ does not contain critical points, then $X_H^t|_{\mathcal{E}^\star_{H,e}}$ is Anosov; if $\mathcal{E}^\star_{H,e}$ has critical points, then there exists $\tilde{e}$, arbitrarily close to $e$, such that $X_H^t|_{\mathcal{E}^\star_{H,\tilde{e}}}$ is Anosov.

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摘要：

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