Research on Perron-Frobenius operator and fractals

Perron-Frobenius算子与分形研究

• 批准号: 14540189
• 负责人: MORI Makoto
• 金额: $ 1.79万
• 依托单位: Nihon 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-14540189/
• 关键词: ergodic theory; dynamical system; Hausdorff dimension; spectrum; psudo random numbe; 乱数; discrepancy; Perron-Frobenius作用素; Perron-Frobenius operator; フラクタル

Expanding the idea of van der Corput sequences, we construct random numbers using inverse images of piecewise linear transformations. The discrepancy of this random number has deep connection with the ergodic properties of the dynamical system generated by piecewise linear transformations. For example, it is proved that if the dynamical system is mixing, then the random number is uniformly distributed. Moreover, we can deeply study the ergodic properties of the dynamical system by The spectra of thePerron-Frobenius operator. In terms of the spectra, the random number is uniformly distributed if 1 is a simple eigenvalue., and no other eigenvalues on the unit circle. The second greatest eigenvalue of the Perron-Frobenius operator is at least the reciprocal of the slope of the transformation in modulus. We proved that the random number is of low discrepancy if the second greatest eigenvalue equals its minimum in modulus. We extend this idea to construct higher dimensional low discrepancy. sequences, and we succeeded to construct two and three dimensional low discrepancy sequences.We also consider the Hausdorff dimensions from the view point of statistical mechanics, and proved it equals' a zero of the pressure. We also calculate the Hausdorff dimension of trees. The articles of these topics are now submitted.

扩展 van der Corput 序列的思想，我们使用分段线性变换的逆图像构造随机数。这个随机数的差异与分段线性变换生成的动力系统的遍历特性有着深刻的联系。例如，证明如果动力系统是混合的，则随机数是均匀分布的。此外，我们还可以通过Perron-Frobenius算子的谱来深入研究动力系统的遍历性质。从谱上看，如果1是简单特征值，并且单位圆上没有其他特征值，则随机数是均匀分布的。 Perron-Frobenius 算子的第二大特征值至少是模变换斜率的倒数。我们证明，如果第二大特征值等于其模最小值，则随机数的差异较小。我们扩展这个想法来构建更高维度的低差异。我们还从统计力学的角度考虑了Hausdorff维数，并证明了它等于压力为零。我们还计算树的豪斯多夫维数。这些主题的文章现已提交。

项目成果

Yoshihiko Yamaura: "A construction of a Lipschitz continuous minimizer of a free boundary problem"Nonlinear Analysis. 54. 1175-1191 (2003)

Yoshihiko Yamaura：“自由边界问题的 Lipschitz 连续最小化器的构造”非线性分析。

森 真: "Construction of tow dimensional low discrepancy sequences"Monte Carlo methods and Application. vol.8No.2. 159-170 (2002)

Makoto Mori：“二维低差异序列的构造”Monte Carlo 方法和应用，第 8 卷第 159-170 卷（2002 年）。

Y.Ichikawa, M.Mori, M.Ohno: "Hausdorff Dimension of a Cantor set on $R^1$"Tokyo J.Math.. vol.26,No.2. 371-390 (2003)

Y.Ichikawa、M.Mori、M.Ohno：“$R^1$ 上康托尔的豪斯多夫维数”Tokyo J.Math.. vol.26，No.2。

V.Berthe, S.Ferenzi, C.Mauduit, A.Siegel編: "Substitutions in Dynamics, Arithmetics and Combinatorics"Springer. 402 (2004)

V.Berthe、S.Ferenzi、C.Mauduit、A.Siegel 编辑：“动力学、算术和组合学中的替代”Springer 402 (2004)。

Yoshihiko Yamaura: "汎関数の近時理論の適用によるリプシッツ連続な最小化関数の構成"京都大学数理解析研究所考究録. 1254. 23-31 (2002)

Yoshihiko Yamaura：“利用最新泛函理论构建 Lipschitz 连续最小化函数”京都大学数学分析研究所研究报告 1254. 23-31 (2002)。