设K⊂ R为由满足强分离条件的相似压缩映射族{hk(x)=akx+bkx,k=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f(x)=γp([0,x]∩K),这里假设 min1≤k≤ N(log pk/log ak)<1< max1≤k≤ N(log pk/logak) 设数ξ和q+β(q)分别由∑k=1N akξ=1和 ∑k=1N pkq akβ(q)=1,β(q)=-1所确定.本文研究集合K中使得函数f(x)的导数不存在的点集,使得函数f(x)的导数为零的点集,及使得函数f(x)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≤k≤ N(log pk/log ak)< 1 < max1≤k≤ N(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
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Key words
self-similar measure /
generalized Cantor function /
non-differentiability point
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参考文献
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脚注
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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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