TY - JOUR
T1 - On dimensions of multitype Moran sets
AU - Liu, Qing Hui
AU - Wen, Zhi Ying
PY - 2005/11
Y1 - 2005/11
N2 - 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 dimH E=s* ≤ s*= dim P E= dimB 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.
AB - 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 dimH E=s* ≤ s*= dim P E= dimB 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.
UR - http://www.scopus.com/inward/record.url?scp=27244434766&partnerID=8YFLogxK
U2 - 10.1017/S0305004105008686
DO - 10.1017/S0305004105008686
M3 - Article
AN - SCOPUS:27244434766
SN - 0305-0041
VL - 139
SP - 541
EP - 553
JO - Mathematical Proceedings of the Cambridge Philosophical Society
JF - Mathematical Proceedings of the Cambridge Philosophical Society
IS - 3
ER -