分形的Kronecker积及其结构特征

Kronecker Products of Fractals and Its Structural Character

为了给一般分形集的Kronecker积的研究提供理论依据,将矩阵的Kronecker乘积首次应用到分形几何上,从而给出欧式空间中2个分形的Kronecker积运算,说明了平面上自仿射集的Kronecker积并不是一个自仿射集,表明其不是一个传统的不变集;研究了直线上由表示系统所生成的分形集与自身的Kronecker积的结构特征,通过自然分布原理给出了直线上该类Kronecker积的Hausdorff维数的上界,并证明了其一定包含一个内部非空的空间.

In order to provide a theoretical basis for the research on the Kronecker product of the general fractal sets,the Kronecker product of matrices to the fractal geometry for the first time is applied,and the Kronecker product operation between two fractals in Euclidean space is discussed. It is explained that the Kronecker product between two self-affine sets on a plane is not a self-affine set,showing that the new set is also not a traditional invariant set. It studies the structure characteristics of the Kronecker product between a fractal set and itself which is generated by representation system in line. Furthermore,the upper bound of the Hausdorff dimension of Kronecker product in line by the natural distribution principle is given. It is also proven that it must contain a non-empty interval.