流形上的正熵测地流
Positve Entropy Geodesic Flows on Nilmanifolds
摘要：
设T是4 × 4实上三角形矩阵的幂零群.本文证明了T上某些左不变黎曼度量的欧拉方程有马蹄形。借助于Melnikov型积分的数值计算，我们还证明了T上亚黎曼卡诺度规的Euler方程具有马蹄形.这使蒙哥马利、夏皮罗和斯托林的一个较早的结果更加尖锐，他们证明了方程在代数上是不可积的。
英文摘要：
Let T be the nilpotent group of 4 x 4 real upper triangular matrices. In this note we show that the Euler equations of certain left-invariant riemannian metrics on T have a horseshoe. We also show, with the aid of a numerical computation of a Melnikov-type integral, that the Euler equations of the sub-riemannian Carnot metric on T has a horseshoe. This sharpens an earlier result of Montgomery, Shapiro and Stolin who had shown that the equations are algebraically non-integrable.
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