TY - JOUR
T1 - An intermediate value property of fractal dimensions of cartesian product
AU - Jiang, Xiaofang
AU - Liu, Qinghui
AU - Wen, Zhiying
N1 - Publisher Copyright:
© 2017 World Scientific Publishing Company.
PY - 2017/12/1
Y1 - 2017/12/1
N2 - 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].
AB - 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].
KW - Cartesian Product
KW - Hausdorff Dimension
KW - Moran Sets
KW - Packing Dimension
UR - http://www.scopus.com/inward/record.url?scp=85030843073&partnerID=8YFLogxK
U2 - 10.1142/S0218348X17500529
DO - 10.1142/S0218348X17500529
M3 - Article
AN - SCOPUS:85030843073
SN - 0218-348X
VL - 25
JO - Fractals
JF - Fractals
IS - 6
M1 - 1750052
ER -