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

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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