测地流的遍历理论研究进展

Ergodic Theory of Geodesic Flow

测地流是动力系统和微分几何交叉领域的重要课题。负曲率和非正曲率流形上的测地流是双曲动力系统和混沌现象最自然的例子,其遍历性质一直是国际研究热点。介绍测地流遍历理论中的研究进展,包括Liouville测度的遍历性和最大熵测度的唯一性。研究认为运用最大熵测度的遍历性质,可以建立素闭测地线个数增长的渐近公式,即素轨道定理,还可以建立万有覆盖空间中球体积增长的渐进公式。素轨道定理是遍历论中的最大熵测度在黎曼几何中的一个深刻应用。

The geodesic flow is an important topic lying in the crossing field of dynamical systems and differential geometry.The geodesic flow on manifolds of negative curvature and nonpositive curvature is the most natural example of hyperbolic dynamical systems and chaotic phenomenon,and its ergodic properties are extensively studied worldwide.In this paper,several new progresses in the ergodic theory of geodesic flows are presented,including the ergodicity of the Liouville measure,and the uniqueness of measure of maximal entropy.Applying the ergodic properties of the measure of maximal entropy,the asymptotic formula for the growth of the number of prime closed geodesics,the so-called prime orbit theorem,as well as the asymptotic formula for the volume growth of the balls in the universal cover can be established.The prime orbit theorem is a deep application of the measure of maximal entropy in ergodic theory to Riemannian geometry.