- Volume 34 Issue 2
DOI QR Code
THE CORRELATION DIMENSION OF GENERALIZED CANTOR-LIKE SETS
- Lee, Mi-Ryeong (Department of Mathematics, Kyungpook National University) ;
- Baek, Hun-Ki (Department of Mathematics Education, Catholic University of Daegu)
- Received : 2012.03.14
- Accepted : 2012.03.29
- Published : 2012.06.25
In the paper, a symbolic construction is considered to define generalized Cantor-like sets. Lower and upper bounds for the correlation dimension of the sets with a regular condition are obtained with respect to a probability Borel measure. Especially, for some special cases of the sets, the exact formulas of the correlation dimension are established and we show that the correlation dimension and the Hausdorff dimension of some of them are the same. Finally, we find a condition which guarantees the positive correlation dimension of the generalized Cantor-like sets.
Supported by : National Research Foundation of Korea(NRF)
- I. S. Baek, Dimensions of weakly convergent deranged Cantor sets, Real Analysis Exchange, 23(2), (1997-98), 689-696.
- H. Baek, Packing dimension and measure of homogeneous Cantor sets, Bulletin of Aust. Math. Soc., 74(2006), 443-448. https://doi.org/10.1017/S000497270004048X
- P. Billingsley, Ergodic Theory and Information, John Wiley & Sons, (1965).
- C. A. Cabrelli, K. E. Hare and U. M. Molter, Sums of Cantor sets, Ergod. Th. & Dynam. Sys. 17 (1997), 1299-1313. https://doi.org/10.1017/S0143385797097678
- S. K. Chang, M. R. Lee and H. H. Lee, Bounds of correlation dimensions for snapshotattractors, Bull. Korean Math. Soc., 41(2) (2004), 327-335. https://doi.org/10.4134/BKMS.2004.41.2.327
- K. Falconer, Techniques in fractal geometry, Mathematical Foundations and Applications, John Wiley & Sons (1997).
- K. Falconer, Generalized dimensions of measures on almost self-affine sets, Nonlinearity, 23(5)(2010), 1047-1070. https://doi.org/10.1088/0951-7715/23/5/002
- P. Grassberger and I. Procaccia, Characterrzation of Strange Attractors, Physical Review Letters, 50(3)(1983), 346-349. https://doi.org/10.1103/PhysRevLett.50.346
- P. Grassberger, I. Procaccia, Measuring the strangeness of strange attractors, Physica D: Nonlinear Phenomena, 9(1-2)(1983), 189-208. https://doi.org/10.1016/0167-2789(83)90298-1
- K. E. Hare and S. Yazdani, Quasi self-similarity and multi-fractal analysis of Cantor measures, Real Analysis Exchange, 27(1), (2001/2002), 287-308.
- H.G.E. Hentschel, I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Physica D: Nonlinear Phenomena, 8(3)(1983), 435-444. https://doi.org/10.1016/0167-2789(83)90235-X
- J. E. Hutchinson, Fractals and self-similarity, Indiana Math. J., 30(1981), 713-747. https://doi.org/10.1512/iumj.1981.30.30055
- M. R. Lee and S. K. Chang, Dimensions for random loosely self-similar sets, Korean J. Math. Sciences, 9(2002), 1-8.
- M. R. Lee and H. H. Lee, Correlation dimensions of Cantor-like sets, Commun. Korean Math. Soc., 18(2)(2003), 281-288. https://doi.org/10.4134/CKMS.2003.18.2.281
- Curt McMullen, The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J., 96(1984), 1-9. https://doi.org/10.1017/S0027763000021085
- J. Myjak, T. Szarek, On the Hausdorff dimension of Cantor-like sets with overlaps, Chaos, Solitons & Fractals, 18(2)(2003), 329-333. https://doi.org/10.1016/S0960-0779(02)00661-6
- E. Ott Chaos in Dynamical Systems, Cambridge Univercity Press, (1993)
- Y. Pesin and H.Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture, Commun. Math. Phys., 182, (1996), 105-153. https://doi.org/10.1007/BF02506387
- C. Q. Qu, H. Rao. and W. Y. Su, Hausdorff measure of homogenous Cantor set, Acta Math. Sinica, English Series, 17(1)(2001), 15-20. https://doi.org/10.1007/s101140000089
- T. D. Sauer and J. A. Yorke, Are the dimensions of a set and its images equal under typical smooth functions ?, Ergod. Th. & Dynam. Sys., 17(1997), 941-956. https://doi.org/10.1017/S0143385797086252
- K. Simon and B. Solomyak, Correlation dimension for self-similar Cantor sets with overlaps, Fund. Math., 155(3)(1998), 293-300.