Counting common perpendicular arcs in negative curvature
Counting common perpendicular arcs in negative curvature
复制标题
计算负曲率的常见垂直弧
DOI：
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发表时间：
2013
影响因子：
0.9
通讯作者：
F. Paulin
中科院分区：
文献类型：
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作者：
Jouni Parkkonen;F. Paulin
令 $D^{-}$ 和 $D^{+}$ 正确地浸入具有收缩负截面曲率的黎曼流形的局部封闭凸子集。利用测地流的混合特性，我们给出一个渐近公式 $t
ightarrow +infty$ 表示从 $D^{-}$ 到 $D^{+}$ 长度至多 $t$ 的公共垂线的数量，并以重数进行计数，并且我们证明了公共垂线端点处的切向量 $D^{-}$ 和 $D^{+}$ 的外部和内部单位法束中的等分布。当流形是紧致的且具有相关性的指数衰减或有限体积的算术时，我们给出渐近的误差项。作为一个应用，我们给出了当克莱因群直径达到 $0$ 时，克莱因群不连续域的连通分量数量的渐近公式。
Let $D^{-}$ and $D^{+}$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t
ightarrow +infty$ for the number of common perpendiculars of length at most $t$ from $D^{-}$ to $D^{+}$ , counted with multiplicities, and we prove the equidistribution in the outer and inner unit normal bundles of $D^{-}$ and $D^{+}$ of the tangent vectors at the endpoints of the common perpendiculars. When the manifold is compact with exponential decay of correlations or arithmetic with finite volume, we give an error term for the asymptotic. As an application, we give an asymptotic formula for the number of connected components of the domain of discontinuity of Kleinian groups as their diameter goes to $0$ .