允许揉搓序列在乘积度量下的维数和测度

张爱华, 陈二才

数学学报 ›› 2009, Vol. 52 ›› Issue (6) : 1207-1210.

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数学学报 ›› 2009, Vol. 52 ›› Issue (6) : 1207-1210. DOI: 10.12386/A2009sxxb0150
论文

允许揉搓序列在乘积度量下的维数和测度

    张爱华, 陈二才
作者信息 +

Dimension and Measure of Admissible Kneading Sequences with Product Metric

    Ai Hua ZHANG, Er Cai CHEN
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摘要

利用Hausdorff维数和Hausdorff测度, 对单峰映射的允许揉搓序列的集合给出定量的刻画, 证明了该集合在乘积度量下的符号空间Σ2中的Hausdorff维数为s, s 维Hausdorff测度为零. 这与传统的定性分析相比, 结果更有意义.

 

Abstract

Using the tools of Hausdorff dimension and Hausdorff measure, we give quantitative version for the set of admission kneading sequences with product metric. It is proved for the set that the Hausdorff dimension is s and the s-dimension Hausdorff measure is zero in Σ2, which are more profound than those obtained by traditional qualitative analysis.

 

关键词

单峰映射 / 允许揉搓序列 / Hausdorff维数和 Hausdorff测度

Key words

unimodal map / admissible kneading sequence / Hausdorff measure and Hausdorff dimension

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张爱华, 陈二才. 允许揉搓序列在乘积度量下的维数和测度. 数学学报, 2009, 52(6): 1207-1210 https://doi.org/10.12386/A2009sxxb0150
Ai Hua ZHANG, Er Cai CHEN. Dimension and Measure of Admissible Kneading Sequences with Product Metric. Acta Mathematica Sinica, Chinese Series, 2009, 52(6): 1207-1210 https://doi.org/10.12386/A2009sxxb0150

参考文献


[1] Collet P., Eckmann J., Iterated Maps of the Intervals as Dynamical Systems, Boston: Birkhauser, 1980.

[2] de Melo W., van Strien S., One-Dimensional Dynamics, Ergebnisse der Mathematik und Ihrer Grenzgebiete (3), Berlin: Springer-Verlag, 1993.

[3] John M., William T., On iterated maps of the interval, Dynamical Systems, Lecture Notes in Math., 1988, 1342: 465--563.

[4] Huang G. F., Wang L. D., Liao G. F., Likely Limit Sets and Hausdorff Dimension of Unimodal Feigenbaum's Maps, Journal of Jilin University (Science Edition), 1999, 37 (1): 4--6 (in Chinese).

[5] Wen Z. Y., Mathematical Foundations of Fractal Geometry, Shenyang: Northeast University Press, 2001 (in Chinese).

[6] Zhang A. H., Liao G. F., Yan Z. Z., Dimension and measure of unimodal maps' admissible kneading sequences, Chinese Annals of Mathematics, 2006, 27(2): 255--258 (in Chinese).

基金

国家自然科学基金资助项目(10571086)

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