Hausdorff Dimension Of Some Reduced Homogeneous Moran Sets

The Moran sets, a class of typical fractal sets, are very interesting to the researcher in recent years. For the complex of Moran sets, the works of them is still focus on homogeneous Moran sets at present. The Hausdorff dimension and the packing dimension of homogenous Moran sets are given when the lower boundary of the rate of compression is greater than zero. The Hausdorff dimension of higher dimensional homogenous Moran sets and the product sets of homogenous Moran sets are developed also.The sets that are constructed by reduction of k-sequence sets in the iterative process of the homogeneous Moran sets are called the reduced homogeneous Moran sets. Because they are not homogeneous Moran sets, we can not study them by mass distribution or decreasing order and additional basic units, we can not use the way of Cantor sets to prove equivalent cover net yet. But some homogeneity of the reduced homogeneous Moran sets is still held, the Hausdorff dimension of them can be given by some methods. So the investigation of reduced homogeneous Moran sets is very important to the research of general Moran sets.The main works of this paper are followed.â… . The origin, development and current research situation of Moran sets are reviewed, and the background of research is introduced. Then the definition, properties and general research methods of the Hausdorff dimension are listed. The construction, definition and properties of the Moran sets are given. Based on it, reduced homogeneous Moran set is defined.â…¡. The reduced homogeneous Moran set (E|⁾ that are constructed by reducing k-sequence sets in the iterative process of the homogeneous Moransets to(D|⁾_k={(i₁,…,i_k):1≤i_?≤n_?,i_?≠2 and i_?≠3 unless i_(?)-1=1,2≤j≤k}.The Hausdorff dimension of it is gained by analyzing the numbers and increasing or decreasing orders regularity of basic units wheren₁=n₃-2=n₅-2 =…= n_2k+1-2=m₁, n₂-2=n₄-2=…= n_2k-2=m₂ and m₁2. The Hausdorff dimension of (E|⁾ isâ…¢. The reduced homogeneous Moran set E that are constructed by reducing k-sequence sets in the iterative process of the homogeneous Moran sets to(D|⁾_k={(i₁,…,i_k)∈N^k:1≤i_?≤n_?,i_?≠2as i_(?)-1=1,2≤j≤k}. The Hausdorff dimension of it is gained by analyzing the numbers andincreasing or decreasing orders regularity of basic units wheren₁=n₃-1=n₅-1=…= n_2k+1-1=m₁, n₂-1=n₄-1=…=n_2k-1=m₂ and m₁2.The Hausdorff dimension of (E|⁾ is...