基于散射的零流形上的正熵Hamilton系统
Positive-entropy Hamiltonian systems on Nilmanifolds via Scattering
摘要：
令$Σ$是$T_4$的紧商，$T_4$是沿对角线具有单位的$4 \times 4$上三角矩阵的李群。$T_4$的李代数$t_4$具有矩阵的标准基$\{X_{ij}\}$，其中除了在$(i,j)$条目中之外处处具有$0$，这是统一的。令$g$是卡诺度量，一个亚黎曼度量，在$T_4$上，其中$X_{i,i+1}$，$(i=1,2,3)$是正交基。Montgomery、Shapiro和Stolin证明了$g$的测地线流是代数不可积的。本文证明了$T Σ$上该卡诺度量的测地线流具有正的拓扑熵，并且是实解析不可积的.它扩展了Butler和Gelfreich的早期工作。
英文摘要：
Let $Σ$ be a compact quotient of $T_4$, the Lie group of $4 \times 4$ upper triangular matrices with unity along the diagonal. The Lie algebra $t_4$ of $T_4$ has the standard basis $\{X_{ij}\}$ of matrices with $0$ everywhere but in the $(i,j)$ entry, which is unity. Let $g$ be the Carnot metric, a sub-riemannian metric, on $T_4$ for which $X_{i,i+1}$, $(i=1,2,3)$, is an orthonormal basis. Montgomery, Shapiro and Stolin showed that the geodesic flow of $g$ is algebraically non-integrable. This note proves that the geodesic flow of that Carnot metric on $T Σ$ has positive topological entropy and is real-analytically non-integrable. It extends earlier work by Butler and Gelfreich.
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