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

我们注意到复上半平面、模群、模曲面分别被认为是亏格为1的曲线的Teichm011er空间、映射类群和模空间。因此，考虑亏格大于1的情形似乎是很自然的。在亏格1的情况下，众所周知，模曲面的闭测地线、本原双曲元素的共轭类以及线性分式变换的二重迭代的周期轨道是自然一一对应的，闭测地线的分布满足素数型定理。这项工作的目的是建立双曲曲线模空间的同类结果，特别是与重正化Rauzy诱导的周期轨道相对应的一类受限Teichmtiller闭测地线的素数型定理。2004年，我们试图得到动力Zeta f…的亚纯扩张重整化Rauzy诱导的更多函数。我们需要建立一种系统的方法来在足够宽的半平面上扩展具有奇性的双曲动力学的动力Zeta函数。为此，我们首先处理一些典型的例子，以找到这些动力学的共同结构。作为结果，我们通过构造Lipschitz连续不变叶，得到了将无食二维散射开放台球的Zeta函数亚纯推广到含有多个非负实部组成的半平面的区域的方法。2005年，我们将2004年得到的上述结果推广到一类具有Cantor不变集的Lipschitz连续马尔可夫系统，并建立了相应的动态Zeta函数在原点的特殊值的计算方法。另外，我们还证明了一类重整化Rauzy诱导的弱型局部极限定理。2006年，我们注意到Bufetov最近的结果可以用来证明一类重整化Rauzy-Veech-Zorich诱导的素数型定理，这类重整化Rauzy-Veech-Zorich诱导是一类有趣但更具限制性的重整化Rauzy诱导。包含这一结果的论文将在不久的将来提交给相应的期刊。较少

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