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

点击翻译按钮获取中文摘要

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.