Ergodic geometry for non-elementary rank one manifolds
Ergodic geometry for non-elementary rank one manifolds
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DOI：
10.3934/dcds.2016072
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发表时间：
2015-08
期刊：
影响因子：
--
通讯作者：
G. Link;J. Picaud
中科院分区：
文献类型：
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作者：
G. Link;J. Picaud
设$X$是一个Hadamard流形，$\Gamma$是$X$的非初等离散等距群，其中包含一个秩为1的等距。我们将商orbifold $M=X/\Gamma$的测地线流的遍历理论与$\Gamma$的庞加莱级数的行为联系起来。准确地说，本文的目的是将所谓的Hopf-Tsuji-Sullivan定理-众所周知的流形的捏负曲率-秩一orbifolds的框架。此外，我们还得到了$\Gamma$的几何极限集上的$\Gamma$-不变共形密度的一些重要性质。
Let $X$ be a Hadamard manifold, and $\Gamma$ a non-elementary discrete group of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/\Gamma$ to the behavior of the Poincar{\'e} series of $\Gamma$. Precisely, the aim of this paper is to extend the so-called theorem of Hopf-Tsuji-Sullivan -- well-known for manifolds of pinched negative curvature -- to the framework of rank one orbifolds. Moreover, we derive some important properties for $\Gamma$-invariant conformal densities supported on the geometric limit set of $\Gamma$.