arXiv 1111.20602011-11math.DSmath.GT

负曲面上测地线自相交计数的统计估计

Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces

Steven P. Lalley

发表信息：: Duke Math. J. 163, no. 6 (2014), 1191-1261
DOI：: 10.1215/00127094-2649425

摘要：

设$Υ$为紧致负曲面。从$Υ$上所有长度为$\leq L$的闭测地线的（有限）集合中，随机选择一条，比如$γ_{L}$，并设$N (γ_{L})$为其自相交的个数。已知存在正常数$κ$，其取决于度量，使得$N (γ_{L})/L^{2} \rightarrow κ$在概率上为$L\rightarrow \infty$。本文的主要结果涉及$N (γ_{L})$关于$κL^{2}$的典型涨落的大小。证明了当度量具有常曲率-1时，典型的涨落是$L$阶的，特别是$(N (γ_{L})-κL^{2})/L$弱收敛于一个非退化概率分布.与此相反，还证明了如果度量具有可变的负曲率，则$N (γ_{L})$的涨落为$L^{3/2}$阶，特别地，$(N (γ_{L})-κL^{2})/L^{3/2}$弱收敛于高斯分布.对于一般测地线，即初始切向量根据归一化刘维尔测度随机选取的测地线，也证明了类似的结果。

英文摘要：

Let $Υ$ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $Υ$ of length $\leq L$, choose one, say $γ_{L}$, at random and let $N (γ_{L})$ be the number of its self-intersections. It is known that there is a positive constant $κ$ depending on the metric such that $N (γ_{L})/L^{2} \rightarrow κ$ in probability as $L\rightarrow \infty$. The main results of this paper concern the size of typical fluctuations of $N (γ_{L})$ about $κL^{2}$. It is proved that if the metric has constant curvature -1 then typical fluctuations are of order $L$, in particular, $(N (γ_{L})-κL^{2})/L$ converges weakly to a nondegenerate probability distribution. In contrast, it is also proved that if the metric has variable negative curvature then fluctuations of $N (γ_{L})$ are of order $L^{3/2}$, in particular, $(N (γ_{L})-κL^{2})/L^{3/2}$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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