An intermediate value property of fractal dimensions of cartesian product

Xiaofang Jiang, Qinghui Liu*, Zhiying Wen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

Given two metric spaces E,F, it is well known that, dimHE +dimHF ≤ dimH(E × F) ≤dimHE +dimPF, dimHE +dimPF ≤ dimP(E × F) ≤dimPE +dimPF, where dimHE, dimPE denote, respectively, the Hausdorff and packing dimension of E. In this paper, we show that, for any 0 ≤ s,t ≤ 1, there exist E,F a" such that the following equalities hold simultaneously: dimH(E × F)-dimHE-dimHF = s, dimPE +dimPF-dimP(E × F) = t. This complete the related results of Wei et al. [C. Wei, S. Y. Wen and Z. X. Wen, Remarks on dimensions of Cartesian product sets, Fractals 24(3) (2016) 1650031].

Original languageEnglish
Article number1750052
JournalFractals
Volume25
Issue number6
DOIs
Publication statusPublished - 1 Dec 2017

Keywords

  • Cartesian Product
  • Hausdorff Dimension
  • Moran Sets
  • Packing Dimension

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Jiang, X., Liu, Q., & Wen, Z. (2017). An intermediate value property of fractal dimensions of cartesian product. Fractals, 25(6), Article 1750052. https://doi.org/10.1142/S0218348X17500529