Project/Area Number  10640105 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Tokyo Institute of Technology 
Principal Investigator 
MORITA Takehiko Tokyo Institute of Technology, Graduate School of Science and Engineering, AP, 大学院・理工学研究科, 助教授 (00192782)

CoInvestigator(Kenkyūbuntansha) 
NAKADA Hitoshi Keio University, Faculty of Science and Engineering, AP, 理工学部, 助教授 (40118980)
SHIRAI Tomoyuki Tokyo Institute of Technology, Graduate School of Science and Engineering, A, 大学院・理工学研究科, 助手 (70302932)
伊藤 秀一 東京工業大学, 大学院・理工学研究科, 助教授 (90159905)
UCHIYAMA Kohei Tokyo Institute of Technology, Graduate School of Science and Engineering, P, 大学院・理工学研究科, 教授 (00117566)
SHIGA Tokuzo Tokyo Institute of Technology, Graduate School of Science and Engineering, P, 大学院・理工学研究科, 教授 (60025418)
SHIGA Hiroshige Tokyo Institute of Technology, Graduate School of Science and Engineering, P, 大学院・理工学研究科, 教授 (10154189)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,300,000 (Direct Cost : ¥3,300,000)
Fiscal Year 1999 : ¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 1998 : ¥1,800,000 (Direct Cost : ¥1,800,000)

Keywords  geodesic flow / mapping class group / ergodic theory / thermodynamic formalism / 測地流 / 写像類群 / エルゴード理論 / 熱力学形式 / 閉測地線 
Research Abstract 
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 twodimensional scattering billiards.
