广义Cantor函数的不可微点集的维数

In Soo BAEK, 李文侠

数学学报 ›› 2014, Vol. 57 ›› Issue (5) : 939-946.

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数学学报 ›› 2014, Vol. 57 ›› Issue (5) : 939-946. DOI: 10.12386/A2014sxxb0087
论文

广义Cantor函数的不可微点集的维数

    In Soo BAEK1, 李文侠2
作者信息 +

Dimensions of Non-differentiability Points of Generalized Cantor Functions

    In Soo BAEK1, Wen Xia LI2
Author information +
文章历史 +

摘要

K⊂ R为由满足强分离条件的相似压缩映射族{hkx)=akx+bkxk=1,...,N} 所生成的自相似集,此处N≥2. 对一个概率向量 p=(p1,...,pN),设γp为对应的支撑在K上的自相似测度.在单位线段上定义广义Cantor函数fx)=γp([0,x]∩K),这里假设 min1≤kN(log pk/log ak)<1< max1≤kN(log pk/logak) 设数ξq+βq)分别由∑k=1N akξ=1和 ∑k=1N pkq akβq=1,βq)=-1所确定.本文研究集合K中使得函数fx)的导数不存在的点集,使得函数fx)的导数为零的点集,及使得函数fx)的导数为无穷的点集的维数,本文结果表明上述定义的两个数可以给出这些维数的一个很好的刻画.

Abstract

For a probability vector p = (p1, . . ., pN), there is a corresponding selfsimilar measure γp supported on the generalized Cantor set K in R generated by the family {hk(x)=akx+bkx,k=1,...,N} (N ≥ 2) of contractive similitudes satisfying the strong separation condition. We consider the generalized Cantor function f(x) = γp([0, x] ∩ K) satisfying min1≤kN(log pk/log ak)< 1 < max1≤kN(log pk/log ak) on the unit interval. The numbers q+β(q) and ξ, where β'(q) = -1 with ∑k=1N pkq akβ(q)=1, and ∑k=1N akξ=1 give full information for the dimensions of the non-differentiability points and the null differentiability points and the infinity differentiability points of K.

关键词

自相似测度 / 广义Cantor函数 / 不可微点

Key words

self-similar measure / generalized Cantor function / non-differentiability point

引用本文

导出引用
In Soo BAEK, 李文侠. 广义Cantor函数的不可微点集的维数. 数学学报, 2014, 57(5): 939-946 https://doi.org/10.12386/A2014sxxb0087
In Soo BAEK, Wen Xia LI. Dimensions of Non-differentiability Points of Generalized Cantor Functions. Acta Mathematica Sinica, Chinese Series, 2014, 57(5): 939-946 https://doi.org/10.12386/A2014sxxb0087

参考文献

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基金

National Research Foundation of Korea Grant(NRF-2012S1A2A1A01028519);国家自然科学基金资助项目(11271137)

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