英文题名：
Dimension of Cantor Function Non-differentiable Point Set
中文关键词：
Hausdorff维数;;Packing维数;;Cantor集;;Cantor函数
中文摘要：
分形几何在近三十年来迅速发展成为一门新兴的数学分支,其理论在众多领域中得到了广泛的应用.1883年德国数学家Cantor提出了现在大家所熟知的Cantor三分集,Cantor三分集的构造是很简单的,然而,它却能体现最典型的分形几何特征,我们可以计算出经典三分Cantor集的Hausdorff维数为log 2/log 3.关于经典三分Cantor集,我们对它的研究不仅仅限于最初的测度计算和维数的证明.实际上很多学者对于Cantor函数不可微点的工作也做了很多深入的研究,在1993年,Darst[24]证明了经典三分Cantor集的Cantor函数不可微点的集合的Hausdorff维数为(log 2/log 3)~2,并提到这一结论可以推广到一般的Cantor函数.Eidswick则指出三分Cantor集的Cantor函数的特征在于三进展式中0和2的间隔,并且证明了Cantor函数不可微点集的一个子集T_(μλ)的势是连续统,该子集T_(μλ)是对微分性质更精细的刻画,为此我们可进一步计算出集合T_(μλ)的Hausdorff维数以及Packing维数.
英文摘要：
Fractal geometry has rapidly developed into a new branch of mathematics in the past three decades,and its theory has been widely used in many fields.In 1883,Cantor,a German mathematician,put forward the Cantor set.The construction of the classic middle-third Cantor set is very simple.However,it can reflect the most typical char-acteristics of fractal geometry,We can calculate the Hausdorff dimension of the classic middle-third Cantor set as log 2/log 3.As for the classic middle-third Cantor set,our re-search is not limited to the initial measure calculation and the proof of dimension.In fact,many scholars have done a lot of in-depth research on the jobs of the non differentiable points of Cantor function in 1993,Darst[24]proves that the Hausdorff dimension of the set of nondifferentiable points of Cantor function of the classic middle-third Cantor set is(log 2/log 3)~2,and points out that this conclusion can be extended to the general Cantor function.Eidswick points out that the Cantor function of the classic middle-third Cantor set is characterized by the interval between 0 and 2 in the three progressive formulas,It is also proved that the potential of a subset T_(μλ)of the set of nondifferentiable points of Cantor function is a continuum,and the subset T_(μλ)is a more precise characterization of differential properties.Therefore,we can further calculate the Hausdorff dimension and packing dimension of the set T_(μλ).
