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

概率论和复分析中出现的经典动力系统的遍历理论方法

• 批准号: 10640105
• 负责人: MORITA Takehiko
• 金额: $ 2.11万
• 依托单位: Tokyo Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640105/
• 关键词: geodesic flow; mapping class group; ergodic theory; thermodynamic formalism; 閉測地線

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.

尽管紧致黎曼曲面上的测地流及其在泰希米勒空间上的类似物（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。这里我们列举了我们获得的一些结果。 Sumi 研究了黎曼球上亚纯函数的随机迭代或偏斜积，并对其 Julia 集的 Hausdorff 维数进行了有效估计。 H.志贺等。等人。得到了无限解析型黎曼曲面的Teichmuller空间上映射类群作用的离散性结果。中田等。等人。引入了与数字的连分数相关的新实数正态性概念，并将该概念与通常的正态性进行了比较。最后受到校长 Morita T. Shiga、K. Uchiyama 和 T. Shirai 教授的概率结果的启发获得了二维散射台球动态 zeta 函数亚纯延拓的结果。

项目成果

K.Uchiyama: "Wtener's test for the random walks with mean zero and finite variance" Ann.Prob.26. 368-376 (1998)

K.Uchiyama：“Wtener 的均值为零和有限方差的随机游走检验”Ann.Prob.26。

J. Aaronson and H. Kakada: "Multiple recurrence of Markov shifts and other infinite measure preserving transformation"to appear in Israel J. Math..

J. Aaronson 和 H. Kakada：“马尔可夫位移的多重递归和其他无限测度保持变换”出现在 Israel J. Math..

C.Kraai Kamp: "On the nomal numbers for continued fractions" Evgod.Th.& Dynan.Sys.to appear.

C.Kraai Kamp：“关于连分数的正规数”Evgod.Th。

T.Shirai: "A trace formula for discrets Schrodinger operators" Publ.RIMS Kyoto Univ.34. 27-41 (1998)

T.Shirai：“离散薛定谔算子的迹公式”Publ.RIMS 京都大学 34。

Z.Li: "A reversibility problem for Fleming-Viot processes"Electronic Communication in Probability. 4. 71-82 (1999)

Z.Li：“弗莱明-维奥特过程的可逆性问题”电子通信概率。