Dimensions of multitype Moran sets with lower limit of the contractions being zero

Abstract

Multitype Moran sets were studied by Liu and Wen (2005) [9]. Under the assumption of primitivity they proved that for any multitype Moran set E with a positive lower bounded condition on contracting ratios,dimH⁡E=s⁎≤s⁎=dimP⁡E=dim‾BE, where s⁎ and s⁎ are the lower and upper pre-dimension according to the natural coverings. In this paper we permit that the lower limit of contraction ratios is zero, and study the two classes of them. For one of the two, by assuming some stronger condition than primitivity, we show that the above formula still holds. For the other, under no assumption of primitivity we prove that dimH⁡E=s⁎ (resp. dim‾BE=s⁎) by assuming a mild condition, and show that the above formula also holds provided primitivity and these two mild conditions are satisfied.