On dimensions of multitype Moran sets

Qing Hui Liu*, Zhi Ying Wen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)541-553
Number of pages13
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume139
Issue number3
DOIs
Publication statusPublished - Nov 2005

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Liu, Q. H., & Wen, Z. Y. (2005). On dimensions of multitype Moran sets. Mathematical Proceedings of the Cambridge Philosophical Society, 139(3), 541-553. https://doi.org/10.1017/S0305004105008686