超整函数族Julia集的Hausdorff维数

杨存基

数学学报 ›› 2010, Vol. 53 ›› Issue (1) : 187-198.

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数学学报 ›› 2010, Vol. 53 ›› Issue (1) : 187-198. DOI: 10.12386/A2010sxxb0023
论文

超整函数族Julia集的Hausdorff维数

    杨存基
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Hausdorff Dimensions of Julia Sets of Families of Transcendental Entire Functions

    Cun Ji YANG
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文章历史 +

摘要

Stallard曾经用一族特殊的整函数说明了:超越整函数的Julia集的Hausdorff维数可以无限接近1.本文证明了该函数族的随机迭代的Julia集的Hausdorff维数也可无限接近于1.另一方面,对任意自然数M及任意实数d∈(1,2),本文给出了M个元素的整函数族其随机迭代的Julia集的Hausdorff维数等于d.

 

Abstract

By a family of transcendental entire functions, Stallard shows that the Hausdorff dimensions of Julia sets of those functions have greatest lower bound equal to one.We prove that the Hausdorff dimensions of Julia sets of two families of transcendental entire functions have greatest lower bound equal to one. On the other hand, for any d∈(1, 2), we prove that there exists a family of transcendental entire functions with Hausdorff dimension equal to d.

 

关键词

函数族 / 随机迭代 / Julia集 / Hausdorf维数

Key words

family of functions / random dynamical system / Julia sets / Hausdorf dimension

引用本文

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杨存基. 超整函数族Julia集的Hausdorff维数. 数学学报, 2010, 53(1): 187-198 https://doi.org/10.12386/A2010sxxb0023
Cun Ji YANG. Hausdorff Dimensions of Julia Sets of Families of Transcendental Entire Functions. Acta Mathematica Sinica, Chinese Series, 2010, 53(1): 187-198 https://doi.org/10.12386/A2010sxxb0023

参考文献


[1] Beardon A. F., Iteration of Rational Functions, Berlin: Springer, 1991.

[2] Carleson L., Gamelin T. W., Complex Dynamics, New York, Berlin: Springer-Verlag, 1993.

[3] Milnor J., Dynamics in One Complex Variable, Introductory Lectures 2nd Edition, Berlin: Vieweg, 2000.

[4] Ren F. Y., Complex Analysis Dynamics, Shanghai: Fu Dan university Press, 1996 (in Chinese).

[5] Zheng J. H., Dynamics of Meromorphic Functions, Beijing: Tsinghua University Press, 2006 (in Chinese).

[6] Baker I. N., The domains of normality of an entire function, Ann. Acad. Sci. Fenn. Ser. A. I Math., 1975, 1: 277--283.

[7] McMullen C., Area and hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 1987, 300: 329--342.

[8] Stallard G. M., The Hausdorff dimension of Julia sets of entire functions, Ergod. Th. and Dynam. Sys., 1991, 11: 769--777.

[9] Stallard G. M., The Hausdorff dimension of Julia sets of entire functions II, Math. Proc. Camb. Phil. Soc., 1996, 119: 513--536.

[10] Stallard G. M., The Hausdorff dimension of Julia sets of entire functions III, Math. Proc. Camb. Phil. Soc., 1997, 122: 223--244.

[11] Stallard G. M., The Hausdorff dimension of Julia sets of entire functions IV, J. London Math. Soc., 2000, 61(2): 471--488.

[12] Pólya G., Szegö G., Problems and Theorems in Analysis I (part III, problem 158--160), New York: Springer, 1972.

[13] Zhou W. M., Ren F. Y., The Julia set of the random iteration of transcendental functions, Chinese Science Bull, 1993, 38(2): 249--250.

[14] Duren P. L., Univalent Functions, New York, Berlin, Tokyo: Springer-Verlag, 1983.

基金

国家自然科学基金资助项目(10801134)

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