arXiv 0903.39692009-03math.DS

Anosov Hamilton结构的稳定性

Stability of Anosov Hamiltonian Structures

Will J. Merry, Gabriel P. Paternain

备注：: V2 - minor changes
发表信息：: J. Topol. Anal. 2 (2010) 419--451

摘要：

考虑维数为$n\geq3$的黎曼流形$(M,g)$的切丛，该流形具有负曲率度量（不一定等于$g$），该度量具有由$M$上的闭2-形式定义的扭曲辛结构。我们考虑由标准动能哈密顿量产生的哈密顿流（相对于辛结构），并且我们考虑$H$的紧凑正则能级$Σ_{k}:=H^{-1}(k)$。假设$Σ_{k}$是一个Anosov能级。证明了若$n$是奇的，则若限制在$Σ_{k}$上的Hamilton流是带有$C^{1}$弱丛的Anosov，则Hamilton结构$(Σ_{k}$是稳定的当且仅当它是接触的.如果$n$是偶数，并且假定流动是1/2箍缩的，则同样的结论成立。作为推论，我们推断，如果$g$是负弯曲的，严格1/4-捏和2-形式定义的扭曲辛结构是不准确的，那么哈密顿结构$(Σ_{k}$是永远不稳定的所有足够大的$k$。

英文摘要：

Consider the tangent bundle of a Riemannian manifold $(M,g)$ of dimension $n\geq3$ admitting a metric of negative curvature (not necessarily equal to $g$) endowed with a twisted symplectic structure defined by a closed 2-form on $M$. We consider the Hamiltonian flow generated (with respect to that symplectic structure) by the standard kinetic energy Hamiltonian, and we consider a compact regular energy level $Σ_{k}:=H^{-1}(k)$ of $H$. Suppose $Σ_{k}$ is an Anosov energy level. We prove that if $n$ is odd, then if the Hamiltonian flow restricted to $Σ_{k}$ is Anosov with $C^{1}$ weak bundles then the Hamiltonian structure $(Σ_{k}$ is stable if and only if it is contact. If $n$ is even and in addition the flow is assumed to be 1/2-pinched then the same conclusion holds. As a corollary we deduce that if $g$ is negatively curved, strictly 1/4-pinched and the 2-form defining the twisted symplectic structure is not exact then the Hamiltonian structure $(Σ_{k}$ is never stable for all sufficiently large $k$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文