Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces
Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces
复制标题
负曲面测地线自交计数统计规律
DOI：
--
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
S. Lalley
中科院分区：
文献类型：
--
作者：
S. Lalley
设Upsilon是一个紧致的负曲面。从$Upsilon$上长度为$leq L$的所有封闭测地线的（有限）集合中，随机选择一个，例如$gamma_{L}$，设$N (gamma_{L})$为其自交的个数。已知有一个正常数$kappa$依赖于度规，使得$N (gamma_{L})/L^{2} right - row kappa$在概率上等于$L
right - row inty $。本文的主要结果涉及$N (gamma_{L})$约$kappa L^{2}$的典型波动的大小。证明了当度规具有常曲率-1时，典型波动为$L$阶，特别是$(N (gamma_{L})-kappa L^{2})/L$弱收敛于非退化概率分布。同时证明了如果度规具有可变的负曲率，则$N (gamma_{L})$的涨落为$L^{3/2}$阶，特别是$(N (gamma_{L})-kappa L^{2})/L^{3/2}$弱收敛于高斯分布。对于一般测地线，即根据归一化刘维尔测度随机选取初始切向量的测地线，证明了类似的结果。
Let $Upsilon $ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $Upsilon$ of length $leq L$, choose one, say $gamma_{L}$, at random and let $N (gamma_{L})$ be the number of its self-intersections. It is known that there is a positive constant $kappa$ depending on the metric such that $N (gamma_{L})/L^{2}
ightarrow kappa$ in probability as $L
ightarrow infty$. The main results of this paper concern the size of typical fluctuations of $N (gamma_{L})$ about $kappa L^{2}$. It is proved that if the metric has constant curvature -1 then typical fluctuations are of order $L$, in particular, $(N (gamma_{L})-kappa L^{2})/L$ converges weakly to a nondegenerate probability distribution. In contrast, it is also proved that if the metric has variable negative curvature then fluctuations of $N (gamma_{L})$ are of order $L^{3/2}$, in particular, $(N (gamma_{L})-kappa L^{2})/L^{3/2}$ converges weakly to a Gaussian distribution. Similar results are proved for generic geodesics, that is, geodesics whose initial tangent vectors are chosen randomly according to normalized Liouville measure.