Magnetic flows and Gaussian thermostats on manifolds of negative curvature
Magnetic flows and Gaussian thermostats on manifolds of negative curvature
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负曲率流形上的磁流和高斯恒温器
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发表时间：
2000
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通讯作者：
M. Wojtkowski
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作者：
M. Wojtkowski
我们考虑一类流，其中包括磁场和高斯恒温器的外部领域。我们给出了负截面曲率流形上的这种流动是Anosov的充分条件。0.导论.黎曼流形上的测地线流描述了被限制在流形上的点粒子的惯性运动。如果流形具有负截面曲率，我们得到一个Anosov流，一个具有良好统计性质的动力系统的一个主要例子。在本文中，我们研究由特殊力，磁流和高斯恒温器产生的流动。30年前Anosov和Sinai [A-S]已经讨论了这种背景下的磁通。它们最近由豪达[Go]、Grognet [Gr]和M.和P. Paternain [P-P]。就像测地线流一样，磁通自然地存在于单位切丛上。单位切丛上的另一类流动是最近在物理学文献中引入的，即外场的高斯恒温器[H]。我们证明（第1节），这两类流都可以表示为一般构造的特殊情况。我们定义切丛TM（或余切丛T <$M）上的广义磁流，要求其速度向量场F满足（0.1）iF（ω − γ）= −dH，其中ω是标准辛形式，H是哈密顿量，2-形式γ表示广义磁场。2000年数学学科分类：37 D20、37 D40。我感谢François Ledrappier、Feliks Przytycki、Marek Rychlik和Lai-Sang Young，他们慷慨地向我介绍了他们对SRB措施的了解。
We consider a class of flows which includes both magnetic flows and Gaussian thermostats of external fields. We give sufficient conditions for such flows on manifolds of negative sectional curvature to be Anosov. 0. Introduction. The geodesic flow on a Riemannian manifold describes inertial motion of a point particle confined to the manifold. If the manifold has negative sectional curvature we obtain an Anosov flow, a prime example of a dynamical system with good statistical properties. In the present paper we study flows generated by special forces, magnetic flows and Gaussian thermostats. Magnetic flows in this context were discussed already 30 years ago by Anosov and Sinai [A-S]. They were studied recently by Gouda [Go], Grognet [Gr], and M. and P. Paternain [P-P]. Just like the geodesic flow the magnetic flow lives naturally on the unit tangent bundle. Another class of flows on the unit tangent bundle was introduced recently in physics literature, the Gaussian thermostat of an external field [H]. We show (Section 1) that both classes of flows can be represented as special cases of a general construction. We define a generalized magnetic flow on the tangent bundle TM (or the cotangent bundle T ∗M) by requiring that its velocity vector field F satisfies (0.1) iF (ω − γ) = −dH, where ω is the standard symplectic form, H is a hamiltonian and the 2-form γ represents the generalized magnetic field. 2000 Mathematics Subject Classification: 37D20, 37D40. I thank François Ledrappier, Feliks Przytycki, Marek Rychlik and Lai-Sang Young who generously shared with me their knowledge of SRB measures.