|
|
|
Assouad dimensions of Moran sets and Cantor-like sets |
Wenwen LI1,2,Wenxia LI3,4,Junjie MIAO3,Lifeng XI1,*( ) |
1. Department of Mathematics, Ningbo University, Ningbo 315211, China
2. School of Mathematics and Statistics, Fuyang Normal University, Fuyang 236037, China
3. Department of Mathematics, East China Normal University, Shanghai 200241, China
4. Shanghai Key Lab of PMMP, East China Normal University, Shanghai 200241, China |
|
|
|
|
Abstract We obtain the Assouad dimensions of Moran sets under suitable condition. Using the homogeneous set introduced in [J. Math. Anal. Appl., 2015, 432:888–917], we also study the Assouad dimensions of Cantor-like sets.
|
| Keywords
Fractal
Assouad dimension
Moran set
Cantor-like set
|
|
Corresponding Author(s):
Lifeng XI
|
|
Issue Date: 17 May 2016
|
|
| 1 |
Assouad P. Espaces métriques, plongements, facteurs. Thèse de doctorat, Publ Math Orsay, No 223-7769. Orsay: Univ Paris XI, 1977
|
| 2 |
Assouad P. Ètude d’une dimension métrique liée à la possibilité de plongements dans Rn.C R Acad Sci Paris Sér A-B, 1979, 288(15): 731–734
|
| 3 |
Assouad P. Pseudodistances, facteurs et dimension métrique. In: Seminaire D’Analyse Harmonique (1979-1980). Publ Math Orsay, 80, 7. Orsay: Univ Paris XI, 1980, 1–33
|
| 4 |
Bedford T. Crinkly Curves, Markov Partitions and Box Dimension in Self-similar Sets. Ph D Thesis. Coventry: University of Warwick, 1984
|
| 5 |
Falconer K J. Fractal Geometry—Mathematical Foundations and Applications. Chichester: John Wiley & Sons, Ltd, 1990
|
| 6 |
Fraser J M. Assouad type dimensions and homogeneity of fractals. Trans Amer Math Soc, 2014, 366(12): 6687–6733
https://doi.org/10.1090/S0002-9947-2014-06202-8
|
| 7 |
Heinonen J. Lectures on Analysis on Metric Spaces. New York: Springer-Verlag, 2001
https://doi.org/10.1007/978-1-4613-0131-8
|
| 8 |
Hua S. On the Hausdorff dimension of generalized self-similar sets. Acta Math Appl Sin, 1994, 17(4): 551–558 (in Chinese)
|
| 9 |
Hua S, Li W X. Packing dimension of generalized Moran sets. Prog Nat Sci, 1996, 6(2): 148–152
|
| 10 |
Jin R. Nonstandard methods for upper Banach density problems. J Number Theory, 2001, 91: 20–38
https://doi.org/10.1006/jnth.2001.2671
|
| 11 |
Lalley S, Gatzouras D. Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ Math J, 1992, 41(2): 533–568
https://doi.org/10.1512/iumj.1992.41.41031
|
| 12 |
Li J J. Assouad dimensions of Moran sets. C R Math Acad Sci Paris, 2013, 351(1-2): 19–22
https://doi.org/10.1016/j.crma.2013.01.010
|
| 13 |
Luukkainen J. Assouad dimension: Antifractal metrization, porous sets, and homogeneous measures. J Korean Math Soc, 1998, 35: 23–76
|
| 14 |
Lü F, Lou M L, Wen Z Y, Xi L F. Bilipschitz embedding of homogeneous fractals. J Math Anal Appl, 2015, 432: 888–917
https://doi.org/10.1016/j.jmaa.2015.07.006
|
| 15 |
Mackay J M. Assouad dimension of self-affine carpets. Conform Geom Dyn, 2011, 15: 177–187
https://doi.org/10.1090/S1088-4173-2011-00232-3
|
| 16 |
Mattila P. Geometry of Sets and Measure in Euclidean Spaces. Cambridge: Cambridge University Press, 1995
https://doi.org/10.1017/CBO9780511623813
|
| 17 |
McMullen C. The Hausdorff dimension of general Sierpinski carpets. Nagoya Math J, 1984, 96: 1–9
|
| 18 |
Moran P A. Additive functions of intervals and Hausdorff measure. Math Proc Cambridge Philos Soc, 1946, 42: 15–23
https://doi.org/10.1017/S0305004100022684
|
| 19 |
Olsen L. On the Assouad dimension of graph directed Moran fractals. Fractals, 2011, 19: 221–226
https://doi.org/10.1142/S0218348X11005282
|
| 20 |
Wen Z Y. Mathematical Foundations of Fractal Geometry. Shanghai: Shanghai Scientific and Technological Education Publishing House, 2000 (in Chinese)
|
| 21 |
Wen Z Y. Moran sets and Moran classes. Chinese Sci Bull, 2001, 46: 1849–1856
https://doi.org/10.1007/BF02901155
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
| |
Shared |
|
|
|
|
| |
Discussed |
|
|
|
|
