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首页> 外文期刊>Jahresbericht der Deutschen Mathematiker-Vereinigung >Luis Barreira and Yakov Pesin: 'Introduction to Smooth Ergodic Theory'
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Luis Barreira and Yakov Pesin: 'Introduction to Smooth Ergodic Theory'

机译：路易斯·巴雷拉（Luis Barreira）和雅科夫·佩辛（Yakov Pesin）：“平滑遍历理论简介”

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The main mechanism which causes complicated or chaotic behavior of a deterministic dynamical system is hyperbolic-ity. Hyperbolicity implies that trajectories have some sensitive dependence on initial conditions. Indications of this phenomenon were first discovered by Poincare in his study of the restricted 3-body problem. Not much later first papers on geodesic flows on surfaces of negative curvature appeared by Hadamard, Artin and others. They discovered the complicated dynamics caused by the sign of curvature. In 1939, Hopf showed in a fundamental paper that with respect to the Liouville measure the geodesic flow on compact surfaces of negative curvature is ergodic. The Liouville measure is the smooth measure induced by the canonical symplectic structure of the cotangent bundle. Since the geodesic flow can also be viewed as a Hamiltonian flow, with Hamiltonian function given by the Riemannian metric, the symplectic structure and therefore also the Liouville measure are flow invariant. Ergodicity is a very strong statistical property saying that all flow invariant L~2-functions are constant almost everywhere. It particular, this implies that almost all orbits are dense in phase space. Crucial in Hopf's proof are the stable and unstable foliations of the phase space whose stable resp. unstable leaves consist of manifolds which are exponentially contracted under the geodesic flow in forward resp. backward time. They are also geometrically described by the normal bundle of horospheres where the horospheres are the limits of geodesic spheres on the universal covering. In his proof, Hopf first observed that the time average of a continuous function, which by Birkhoff's ergodic theorem exists almost everywhere, is constant on the stable and unstable manifolds. Using this observation (nowadays called the Hopf argument) in combination with Fubini's theorem, Hopf derived ergodicity of the geodesic flow, by showing that the time averages are constant almost everywhere. However, in order to apply Fubini's theorem some regularity of the stable and unstable foliations is required. For surfaces he obtained a C~1 regularity of the foliations which (for good reasons, see below) he was not able to extend to higher dimensions.
机译：引起确定性动力学系统复杂或混乱行为的主要机制是双曲线性。双曲线意味着轨迹对初始条件有一定的敏感依赖性。庞加莱（Poincare）在研究受限三体问题时首先发现了这种现象的迹象。 Hadamard，Artin等人后来发表的有关负曲率表面上的测地线流动的第一批论文就不多了。他们发现了由曲率符号引起的复杂动力学。 1939年，霍普夫（Hopf）在一份基础论文中表明，对于Liouville量度，负曲率的紧凑表面上的测地线是遍历的。 Liouville测度是由余切束的正则辛结构诱导的平滑测度。由于测地线流也可以看作是哈密顿流，利用黎曼度量给出的哈密顿函数，辛结构以及因此的Liouville度量都是不变的。遍历性是非常强大的统计属性，它表示几乎所有地方的所有流量不变L〜2函数都是恒定的。特别是，这意味着几乎所有的轨道在相空间中都是密集的。霍普夫证明中至关重要的是相空间的稳定和不稳定叶面，其稳定的响应。不稳定的叶子由歧管组成，这些歧管在大地流的作用下呈正向收缩。落后的时间。它们也由法线球体束进行几何描述，其中，球体是测地球在通用覆盖层上的极限。在证明中，霍普夫首先观察到连续函数的时间平均值（在伯克霍夫的遍历定理中几乎存在于各处）在稳定和不稳定流形上都是恒定的。利用这一观察结果（现称为霍普夫定理）结合富比尼定理，霍普夫通过证明时间平均几乎在任何地方都是恒定的，得出了测地流的遍历性。但是，为了应用富比尼定理，需要稳定和不稳定叶片的一些规律性。对于表面，他获得了叶形的C〜1正则性（出于充分的原因，请参阅下文），他无法扩展到更大的尺寸。

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