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，复杂的上半平面是Teichm011er空间的模块化表面，这是映射的类群体，他是1属的曲线空间，因此很自然地考虑大于1的情况。在第1属中，众所周知，模块化表面的封闭测量学，原始C元素的结合类别以及线性分数转换的两倍迭代的周期性轨道是自然的一对一对应关系，并且是尊重的Tyorem的主要类型。更重的rauzy迹象，我们需要建立一种系统的方式，以延长透明度的动力学的动态ZETA函数，并延伸到该条目。通过构造2004年，我们推广了2004年获得的上述结果，从而扩展了带有eclipse的二维散射开放台球的Zeta函数。对于具有类似cantor的不变性集的Lipschitz标记系统，并在相应的动力学Zeta函数上建立了计算的方法，我们还证明了弱类型的限制限制定理。 Bufetov获得的结果可用于证明一类重新归一化的Rauzy-deech-Zorich归纳的质数类型定理，这些介绍很有趣，但更限制的重新分配的rauzy归纳将提交到适当的杂志。在不久的将来