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.

虽然紧致黎曼曲面上的测地线流和Teichmuller空间上的测地线流（两者都是经典哈密顿动力系统的典型例子）都是确定性对象，但众所周知，它们的时间演化看起来像随机现象。我们的研究目的是用热力学方法研究这类确定性经典动力学的内在随机性，这里列举了我们已经得到的一些结果：Sumi教授研究了Riemann球面上亚纯函数的随机迭代或斜积，得到了它们的Julia集的Hausdorff维数的有效估计;滋贺等Nakada等人在无穷解析型Riemann曲面的Teichmuller空间上得到了映射类群作用的离散性结果。等人提出了一个新的与连分式有关的真实的数的正态性概念，并与通常的正态性进行了比较。滋贺，K. Uchiyama和T. Shirai，Morita等得到了二维散射台球的动力学zeta函数的亚纯延拓的一个结果。

项目成果

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..

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。

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。

H. Sumi: "Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products"to appear in Ergod. Th. & Dynam. Sys..

H. Sumi：“亚双曲和半双曲有理半群和偏积的动力学”出现在 Ergod 中。