均匀康托集的Hausdorff中心测度的概率性质

Probability Property of Hausdorff Centred Measure about Uniform Cantor Sets

20世纪90年代C.Tricot给出了Hausdorff中心维数与Hausdorff中心测度的定义,接着人们对分形集的Hausdorff中心维数与Hausdorff中心测度进行研究,结果发现Hausdorff中心测度对测度的重分形谱的估计非常有效 对于均匀康托集K(λ),目前只知Hausdorff中心维数与Hausdorff维数相同 分别借助于数学归纳法和一些细致的不等式估计,给出了均匀康托集K(λ)的概率测度μ(A)=Cs(A∩K(λ))Cs(K(λ))具有不等性质μ([o,r])≤rs,同时构造了K(λ)的一个子集F(λ)满足μ(F(λ))

In 1990s,C.Tricot introduced the notion of Hausdorff centred dimension and Hausdorff centred measure. In the following study on fractal sets by him and others, they found that Hausdorff centred measure is effective in estimating the multifractal spectrum.So far,we only know that the Hausdorff centred dimension is equal to the Hausdorff dimension for the uniform Cantor set K(λ). In this paper, we use some refined estimates of inequality and mathematical induction to obtain the probability property of the uniform Cantor set K(λ),namely,μ()≤rs,where μ()=Cs(A∩K(λ))Cs(K(λ)). At the same time,we construct a set F(λ)K(λ) so that μ(F(λ))=1.