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In this paper we attain the multifractal dimension functions of Moran measure associated with homogeneous Moran fractals and give a sufficient condition which ensures the multifractal spectrum of Moran measure is equal to the Legendre transform of the multifractal dimension functions. As an application of this method, we give an example and a counterexample about the validity of the multifractal formalism of Moran measure.

Recently, Barreira and Schmeling (2000) [1] and Chen and Xiong (1999) [2] have shown, that for self-similar measures satisfying the SSC the set of divergence points typically has the same Hausdorff dimension as the support K . It is natural to ask whether we obtain a similar result for self-similar measures satisfying the OSC. However, with only the OSC satisfied, we cannot do most of the work on a symbolic space and then transfer the results to the subsets of R d , which makes things more difficult. In this paper, by the box-counting principle we show that the set of divergence points has still the same Hausdorff dimension as the support K for self-similar measures satisfying the OSC.