ON INFINITE MEASURE PRESERVING MEASURABLE DYNAMICAL SYSTEMS

关于无限测度保持可测动态系统

• 批准号: 14540214
• 负责人: NAKADA Hitoshi
• 金额: $ 1.86万
• 依托单位: Keio University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540214/
• 关键词: Ergodic Theory; Invariant Measure; 連分数

The following three are the main results of this project:An unite measure preserving transformation can be reduced to a finite measure preserving one as an induced transformation with a set of finite measure. In this case, the ceiling function represents the return time to this set and it is non-integrable. In this sense, we studied the stochastic processes with infinite expectations. In particular, we studied continued fraction mixing stochastic processes with barely infinite expectations. We have proved the strong law of large numbers after light trimming under some conditions on the distribution of random variables.We also studied Maharum extension of ergodic nonsingular transformations. We discussed locally finite invariant measures for Maharam extensions associated to irrational rotations and subshifts. In the case of finite Markov subshifts, we characterized such measures by conformal measures for subshifts. We extend such results to countable Markov subshifts.As an application of infinite ergodic theory, we studied the theory of arithmetic progressions in particular, concerning to Erods conjecture. In this point of view, we considered the multiple recurrence property of infinite measure preserving transformations. We gave an lower estimate of the multiplicity by Kakutani type. We also studied metric number theory as an application of ergodic theory. Some of main results are the following (1) We proved a property of the arithmetic distribution of convergents arising from the Jacobi-Perron algorithm. (2) We constructed Farey maps associated to Rosen's continued fractions. here the Farey maps are 1-dimensional maps which induces mediant convergents of continued fractions.

本文的主要成果有以下三点：一个单位测度保持变换可以化为一个有限测度保持变换，作为一个有限测度集合的诱导变换。在这种情况下，天花板函数表示返回到这个集合的时间，它是不可积的。在这个意义上，我们研究了具有无限期望的随机过程。特别地，我们研究了具有几乎无限期望的连分数混合随机过程。在一定条件下，我们证明了随机变量分布的轻修剪强大数定律。我们还研究了遍历非奇异变换的Maharum扩展。讨论了与非理性旋转和子位移相关的Maharam扩展的局部有限不变测度。在有限马尔可夫子位移的情况下，我们用子位移的保形测度来表征这种测度。我们将这样的结果推广到可数马尔可夫子位移。作为无限遍历理论的一个应用，我们重点研究了等差数列理论，其中涉及到Erods猜想。从这个角度出发，我们考虑了保无穷测度变换的多重递归性。我们对Kakutani类型的多重性给出了较低的估计。我们还研究了度量数论作为遍历理论的一个应用。主要结果如下：(1)证明了由Jacobi-Perron算法引起的收敛性算术分布的一个性质。(2)构建了与Rosen连分式相关的Farey映射。这里的Farey映射是一维映射，它诱导了连分式的中间收敛性。

项目成果

H.Nakada, R.Natsui: "On the metrical theory of continued fraction mixing fibred systems and its application to Jacobi-Perron algorithm"Monatshefte fur Mathematik. 138. 267-288 (2003)

H.Nakada，R.Natsui：“关于连续分数混合纤维系统的度量理论及其在 Jacobi-Perron 算法中的应用”Monatshefte Fur Mathematik。

J.Aaronson, H.Nakada: "Trimmed sums for non negative, mixing stationary processes."Stochastic Process.Appl.. 104. 173-192 (2003)

J.Aaronson、H.Nakada：“非负混合平稳过程的修剪和。”Stochastic Process.Appl.. 104. 173-192 (2003)

G.H.Choe, T.Hamachi, H.Nakada: "Mod 2 normal numbers and Skew products"Studia Math.. (to appear). (2004)

G.H.Choe、T.Hamachi、H.Nakada：“Mod 2 正规数和倾斜产品”Studia Math..（待出现）。

H.Nakada, R.Natsui: "Some strong mixing properties of a sequence of random variables arising from α-continued fractions"Stochastics and Dynamics. 3-4. 463-476 (2003)

H.Nakada、R.Natsui：“α 连续分数产生的随机变量序列的一些强混合特性”随机学与动力学 3-4 (2003)。

H.Nakada, R.Natsui: "Some metric properties of α-continued fractions"Journal of Number Theory. 97. 287-300 (2002)

H.Nakada、R.Natsui：“α-连续分数的一些度量性质”《数论杂志》97. 287-300 (2002)。