欧基里德网格空间中的离散指标

Discrete Indices of Euclidean Lattice Z￣d

在Ｍ．Ｔ．Ｂａｒｌｏｗ和Ｓ．Ｊ．Ｔａｙｌｏｒ研究的基础上，本文定义了欧基里德阿格空间中若干离散分形指标：ｄｉｍ＿Ｌ￣（ｍ）（·），ｄｉｍ＿Ｈ￣（ｍ）（·），δ￣（ｍ）（·），△￣（ｍ）（·）和ｄｉｍ＿ｐ￣（ｍ）（·）．本文研究了这些分形指标的若干内蕴性质，例如ｄｉｍ＿Ｈ￣（ｍ）（·）＝ｄｉｍ＿Ｈ￣（２）（·），δ￣（２￣ｍ）（·）＝δ￣（２）（·）等．为了给Ｚ“中分形子集的研究提供更为便利的理论方法，本文推广了离散情形下的密度引理和Ｆｒｏｓｔ－ｍａｎ引理．本文最后给出了两个例子：例１是由Ｎａｕｄｔｓ提供的；例２来自于位势理论中一个著名的例子．

On the basis of M. T. Barlow and S. J. Taylor, this paper defines several discrete fractal indices of Euclidean Lattice Z￣d, dim_L￣(m),dim_H￣(m),δ￣(m) (· ),△￣(m) (· ) and dim_p￣(m)(·), after givingtwo concepts of m -cube and sin -cube. SOme intrinsic properties of these indices are studied, forexample, dim_H￣(m)(·)= dim_H￣(2) (· ),δ￣(2￣m) (· ) = δ￣(2)(· ) etc.In order to provide more convenient theoretical methods to discuss the fractal subsets ofZ￣d, the discrete analogues of density and Prostman lemmas are generalized.Two examples are given in the end: the first example is due to Naudts ; the second example is based on a well known e-camp:le in potential theory.