Equidistribution of saddle connections on translation surfaces
Equidistribution of saddle connections on translation surfaces
复制标题
平移表面上鞍形连接的均匀分布
DOI：
10.3934/jmd.2019004
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发表时间：
2017
影响因子：
1.1
通讯作者：
B. Dozier
中科院分区：
文献类型：
--
作者：
B. Dozier
固定一个平移曲面$X$，并考虑$X$上的测度，这些测度来自于对所有长度不超过$R$的鞍形联络上的一致测度求平均。则当$R\to\infty$时，这些测度的弱极限存在，并且等于$X$上的Lebesgue测度。我们还证明了由长度不超过$R_n$的所有鞍联络的角所给出的S^1 $上的计数测度的子序列的任何弱极限，如$R_n\to\infty$，都在Lebesgue测度类中.第一个结果的证明使用了第二个结果，以及Kerckhoff-Masur-Smillie的结果，即表面上的定向流几乎在每个方向上都是唯一遍历的。
Fix a translation surface $X$, and consider the measures on $X$ coming from averaging the uniform measures on all the saddle connections of length at most $R$. Then as $R\to\infty$, the weak limit of these measures exists and is equal to the Lebesgue measure on $X$. We also show that any weak limit of a subsequence of the counting measures on $S^1$ given by the angles of all saddle connections of length at most $R_n$, as $R_n\to\infty$, is in the Lebesgue measure class. The proof of the first result uses the second result, together with the result of Kerckhoff-Masur-Smillie that the directional flow on a surface is uniquely ergodic in almost every direction.