In this paper,we generalize a class of multi-type random recursive sets K on structure of sets in Euclidean TV-space constructed by Xu Ciwen, making use of idea which come from Falconer, Mauldin,Williams, Arbeiter, Patzschke etc, and by means of some important methods in matrix analysis and stochastic processes, and obtain the Hausdorff dimensional expression of the multifractal decomposition sets Kα(α> 0) for K on statistically self-similar measures under the strong open set condition.
In Chapter One,we find the relationship between strong open set condition, Hausdorff dimension,the positivity of the Hausdorff measure and β space of graph-directed self-similar sets in complete metric spaces.
Correspondingly, we show that in this case , provided a random strong open set condition is satisfied.
It is well known that the strong open set condition and the open set condition are equivalent for a self-similar set. In this paper we study the property of the strong open set condition. Furthermore, we give an application to local dimension of the self-similar measures.