论文标题
某些无限尺寸分形的平均Hausdorff尺寸
Mean Hausdorff dimension of some infinite dimensional fractals
论文作者
论文摘要
平均Hausdorff维度是Hausdorff维度的动态版本。它提供了一种动力化几何措施理论的方法。我们获得了分形几何形状的以下三个经典结果。 （1）圆圈中均匀集的Hausdorff尺寸的计算。 （2）Hausdorff和Minkowski尺寸的巧合。 （3）计算贝德福德（Bedford）的Hausdorff尺寸 - 麦克穆伦（McMullen）地毯。 我们为平均Hausdorff维度开发了他们的类似物： （1）计算无限尺寸圆环中均质系统的平均值尺寸。 （2）自相似系统的平均Hausdorff维度和度量平均维度的巧合。 （3）无限尺寸地毯的平均豪斯多夫维度的计算。
Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff dimension of homogeneous sets in the circle. (2) The coincidence of Hausdorff and Minkowski dimensions for self-similar sets. (3) The calculation of Hausdorff dimension of Bedford--McMullen carpets. We develop their analogues for mean Hausdorff dimension: (1) The calculation of mean Hausdorff dimension of homogeneous systems in the infinite dimensional torus. (2) The coincidence of mean Hausdorff dimension and metric mean dimension for self-similar systems. (3) The calculation of mean Hausdorff dimension of infinite dimensional carpets.
