Entropy and closed geodesies
Entropy and closed geodesies
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DOI：
10.1017/s0143385700001656
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发表时间：
1982-12
影响因子：
0.9
通讯作者：
A. Katok
中科院分区：
文献类型：
--
作者：
A. Katok
摘要：我们研究在一个具有负截面曲率度量的紧流形上各种黎曼度量下闭测地线的渐近增长。我们的方法利用了测地线的变分和动力学描述，可以被描述为长度 - 面积方法的渐近形式。我们还得到了同一流形上不同度量的测地流的拓扑熵和测度理论熵之间的各种不等式。我们的方法对于任何共形等价于常负曲率度量的度量尤其有效。由于一个经典的正则化定理，对于具有负欧拉特征的曲面，每个黎曼度量都具有此性质。这使我们能够证明，对于足够大的\(T\)，每个非恒定曲率度量在长度至多为\(T\)的闭测地线数量严格多于具有相同总面积的任何常曲率度量。此外，具有固定面积的常负曲率度量的拓扑熵和测度理论熵的共同值将具有相同面积的其他度量的两种熵的值区分开来。
Abstract We study asymptotic growth of closed geodesies for various Riemannian metrics on a compact manifold which carries a metric of negative sectional curvature. Our approach makes use of both variational and dynamical description of geodesies and can be described as an asymptotic version of length-area method. We also obtain various inequalities between topological and measure-theoretic entropies of the geodesic flows for different metrics on the same manifold. Our method works especially well for any metric conformally equivalent to a metric of constant negative curvature. For a surface with negative Euler characteristics every Riemannian metric has this property due to a classical regularization theorem. This allows us to prove that every metric of non-constant curvature has strictly more close geodesies of length at most T for sufficiently large T then any metric of constant curvature of the same total area. In addition the common value of topological and measure-theoretic entropies for metrics of constant negative curvature with the fixed area separates the values of two entropies for other metrics with the same area.