On dimensions of multitype Moran sets

Multitype Moran sets are introduced in this paper. They appear naturally in the study of the structure of the quasi-crystal spectrum, and they generalize some known fractal structures such as self-similar sets, graph-direct sets and Moran sets. It is known that for any Moran set E with a bounded condition on contracting ratios, one has dim_H E=s* ≤ s*= dim _P E= dim_B E, where s* and s* are the lower and upper pre-dimension according to the natural coverings. For any multitype Moran set E with a bounded condition on contracting ratios, we prove dimB E{=}s*. With an additional assumption of primitivity, we prove that for multitype Moran sets, the above formula still holds. We also give some examples to show that if the condition of primitivity is not fulfilled, then it may happen that $\dim_H E or $\dim_P E.