Study Of The Dimension And Dimension Partition Of Cantor Sets

This paper gives some kinds of Cantor sets and their dimensions, and focuses on the study of the Cantor sets C_a defined by monotone non-increasing sequence {a_n}. We can characterize its dimension in terms of the properties of the associated sequence, and get that the Hausdorff dimension is given by dimH ,the pre-packing dimension and packing dimension coincide:dimp upper box dimension is , and the lower box dimension isFor a finer classification, we classify the Cantor sets C_a by the h-Hausdorff measures and h-packing measures, and the classification according to their dimension partitions can be also characterized in terms of their properties. Finally, we get the conclusion that two Cantor sets have the same dimension segmentation when and only when they are associated with dimension functions who are equivalent.