Ergodic theoretical approach to classical dynamical systems appearing in probability theory and complex analysis

尽管紧致黎曼曲面上的测地流及其在Teichmuller空间上的类似流（两者都是经典哈密顿动力系统的典型例子）是确定性对象，但众所周知，它们的时间演化看起来像随机现象。我们项目的目的是通过热力学形式主义来研究这种确定性经典动力学的内在随机性。这里我们列举了一些我们已经得到的结果。Sumi研究了亚纯函数在Riemann球上的随机迭代或偏积，得到了它们的Julia集的Hausdorff维数的一个有效估计。H. Shiga等获得了无限解析型Riemann曲面上映射类群作用在Teichmuller空间上的离散性结果。Nakada等人引入了实数正态性的新概念，该概念与数字的连分数有关，并将其与通常的正态性进行了比较。最后，受T. Shiga， K. Uchiyama和T. Shirai教授的概率结果的启发，Morita主任获得了二维散射台球动态zeta函数亚纯延拓的结果。

Although the geodesic flow on a compact Riemann surface and its analogue on the Teichmuller space (both are typical examples of classical Hamiltonian dynamical systems) are deterministic objects, it is well known that their time evolution look like random phenomena. The aim of our projects is to investigate the intrinsic randomness of such a deterministic classical dynamics by means of thermodynamic formalism.Here we enumerate some results we have obtained.Prof. Sumi studied the random iteration or the skew product of meromorphic functions on the Riemann sphere and obtained an effective estimate for the Hausdorff dimension of their Julia sets.Prof. H. Shiga et. al. obtained the result on discreteness of the action of the mapping class group on the Teichmuller space for Riemann surface of infinite analytic type.Prof. Nakada et. al. introduced a new notion of normality for real numbers which related to the continued fraction of numbers and compare the notion with the usual normality.Finally inspired by the probabilistic results by Professors T. Shiga, K. Uchiyama, and T. Shirai, the head Morita obtained a result on meromorphic continuation of the dynamical zeta function for two-dimensional scattering billiards.