Siegel measures
Siegel measures
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西格尔措施
DOI：
10.2307/121033
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发表时间：
1998
影响因子：
2.7
通讯作者：
W. Veech
中科院分区：
文献类型：
--
作者：
W. Veech
本文的目标首先是描述然后应用数字几何中西格尔积分公式的遍历理论推广。通用公式将被视为既可以作为指导，又可以作为解决有关分布问题的工具，从某种意义上讲，测量叶状结构的闭合叶集服从于闭合黎曼曲面上的亚纯二次微分。在准备讨论主要结果时，我们回顾两个早期的定理。其中第一个由 H. Masur 完成，是当前工作的起点。令 q 为亚纯二次微分，在闭合黎曼曲面 X 上具有最坏的简单极点。对于某个可数集合 θ ∈ R，与 e−2iθq 相关的水平叶理具有一个或多个闭合叶的最大圆柱体。每个圆柱体确定一对向量 v = ±reiθ，其中 r 是圆柱体中闭合叶子的公共 |q|-长度。令 Π(q) 为具有重数的向量集，当 θ 变化时，这些向量由封闭圆柱体产生。最后，令 N(q,R) = Card{v ∈ Π(q) | |v| < R} 是 Π(q) 的增长函数。
The goals of this paper are first to describe and then to apply an ergodictheoretic generalization of the Siegel integral formula from the geometry of numbers. The general formula will be seen to serve both as a guide and as a tool for questions concerning the distribution, in senses to be made precise, of the set of closed leaves of measured foliations subordinate to meromorphic quadratic differentials on closed Riemann surfaces. In preparation of a discussion of the main results we recall two earlier theorems. The first of these, by H. Masur, has been a starting point for the present work. Let q be a meromorphic quadratic differential with at worst simple poles on a closed Riemann surface X. For a certain countable set of θ ∈ R the horizontal foliation associated to e−2iθq has one or more maximal cylinders of closed leaves. Each cylinder determines a pair of vectors v = ±reiθ, where r is the common |q|-length of closed leaves in the cylinder. Let Π(q) be the set of vectors, with multiplicities, which arise from closed cylinders as θ varies. Finally, let N(q,R) = Card{v ∈ Π(q) | |v| < R} be the growth function of Π(q).