On the structures and dimensions of Moran sets

The Moran sets and the Moran class are defined by geometric fashion that distinguishes the classical self-similar sets from the following points: (i) The placements of the basic sets at each step of the constructions can be arbitrary. (ii) The contraction ratios may be different at each step. (iii) The lower limit of the contraction ratios permits zero. The properties of the Moran sets and Moran class are studied, and the Hausdorff, packing and upper Box-counting dimensions of the Moran sets are determined by net measure techniques. It is shown that some important properties of the self-similar sets no longer hold for Moran sets.