权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Ergodic-theoretical study of the distribution of Teichmuller closed geodesics and the dynamical zeta functions

Teichmuller 闭合测地线分布和动态 zeta 函数的遍历理论研究

基本信息

批准号：
16340048
负责人：
MORITA Takehiko
金额：
$ 7.55万
依托单位：
Hiroshima University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16340048/
关键词：
Teichmuller space dynamical zeta function thermodynamic formalism mapping class group ergodic theory

项目摘要

We note that the complex upper half-plane, the modular group, the modular surface are regarded as the Teichm011er space, the mapping class group, and the moduli space of curves of genus 1, respectively. Therefore it seems natural to consider the case of genus greater than 1. In the case of genus 1, it is well known that the closed geodesics of modular surface, the conjugacy classes of primitive hyperbolic elements, and the periodic orbits of two-fold iteration of the linear fractional transformations are in the natural one-to-one correspondence and the distribution of closed geodesics satisfies the prime number type theorem. The purpose of this project is to establish the same kind of results for the moduli spaces of hyperbolic curves, especially the prime number type theorem for a restrictive class of Teichmtiller closed geodesics which corresponding to the periodic orbits of the renormalized Rauzy inductions.In 2004, we try to obtain the meromorphic extensions of the dynamical zeta f … More unctions of the renormalized Rauzy inductions. We need to establish a systematic way to extend the dynamical zeta functions of hyperbolic dynamics with singularity in sufficiently wide half-plane. To this end we first treat some typical examples to find common structure of those dynamics. As a consequence we obtain a method to extends the zeta function of two-dimensional scattering open billiards without eclipse meromorphically to the domain containing the half-plane consisting of numbers of nonnegative real part by constructing the Lipschitz continuous invariant foliations.In 2005 we generalize the above results of the meromorphic extension obtained in 2004 to a class of Lipschitz continuous Markov systems with Cantor-like invariant sets and establish a way to calculate the special values at the origin for the corresponding dynamical zeta functions. In addition, we also prove the weak type local limit theorem for a class of renormalized Rauzy inductionsIn 2006, we notice that the recent result obtained by Bufetov can be applied to prove the prime number type theorem for a class of renormalized Rauzy-Veech-Zorich inductions which are interesting but more restrictive class of the renormalized Rauzy inductions. The paper containing the result will be submitted to the appropriate journal in the near future. Less

我们注意到复上半平面、模群和模曲面分别被看作亏格为1的曲线的Teichm 011 er空间、映射类群和模空间。因此，考虑亏格大于1的情况似乎是自然的。在亏格为1的情形下，模曲面的闭测地线、本原双曲元的共轭类和线性分式变换二重迭代的周期轨道是自然一一对应的，并且闭测地线的分布满足素数型定理.本项目的目的是在双曲曲线的模空间中建立类似的结果，特别是在一类限制性的Teichmtiller闭测地线上建立素数型定理，这类闭测地线对应于重整化Rauzy归纳的周期轨道 ...更多信息重正化Rauzy归纳的函数。我们需要建立一个系统的方法来扩展具有奇异性的双曲动力学的动力学zeta函数在足够宽的半平面。为此，我们首先处理一些典型的例子，找到这些动力学的共同结构。通过构造Lipschitz连续不变叶理，我们得到了将二维无食散射开台球的zeta函数亚纯延拓到包含非负真实的个数的半平面的区域上的方法.在2005年，我们将2004年得到的亚纯延拓结果推广到了一类具有Cantor-像不变集，并建立一种方法来计算相应的动态zeta函数在原点的特殊值。2006年，我们注意到Bufetov的最新结果可以应用于一类重整化Rauzy-Veech-Zorich归纳的素数型定理的证明，这类归纳是一类有趣但限制性更强的重整化Rauzy归纳.载有这一结果的论文将在不久的将来提交给适当的期刊。少