# PACKING DIMENSION OF MEASURES ON A RANDOM CANTOR SET

• Published : 2004.09.01
• 45 4

#### Abstract

Packing dimension of a set is an upper bound for the packing dimensions of measures on the set. Recently the packing dimension of statistically self-similar Cantor set, which has uniform distributions for contraction ratios, was shown to be its Hausdorff dimension. We study the method to find an upper bound of packing dimensions and the upper Renyi dimensions of measures on a statistically quasi-self-similar Cantor set (its packing dimension is still unknown) which has non-uniform distributions of contraction ratios. As results, in some statistically quasi-self-similar Cantor set we show that every probability measure on it has its subset of full measure whose packing dimension is also its Hausdorff dimension almost surely and it has its subset of full measure whose packing dimension is also its Hausdorff dimension almost surely for almost all probability measure on it.

#### Keywords

packing dimension;random Cantor set

#### References

1. I. S. Baek and H. H. Lee, Perturbed type random Cantor set, Real Anal. Exchange 23 (1997), no. 1, 223–234
2. I. S. Baek, Dimensions of weakly convergent deranged Cantor sets, Real Anal. Exchange 23 (1998), no. 2, 689–696
3. I. S. Baek, Weak local dimension on deranged Cantor sets, Real Anal. Exchange 26 (2001), no. 2, 553–558
4. I. S. Baek, Hausdorff dimension of perturbed Cantor sets without some boundedness condition, Acta Math. Hungar. 99 (2003), no. 4, 279–283 https://doi.org/10.1023/A:1024631512342
5. C. D. Cutler and L. Olsen, A variational principle for the Hausdorff dimensions of fractal sets, Math. Scand. 74 (1994), 64–72
6. K. J. Falconer, Fractal geometry, John Wiley and Sons, 1990
7. K. J. Falconer, The multifractal spectrum of statistically self-similar measures, J. Theoret. Probab. 7 (1994), 681–702 https://doi.org/10.1007/BF02213576
8. K. J. Falconer, Techniques in fractal geometry, John Wiley and Sons., 1997
9. H. Joyce and D. Preiss, On the existence of subsets of finite positive packing measure, Mathematika 42 (1995), 15–24 https://doi.org/10.1112/S002557930001130X
10. A. Mucci, Limits for martingale like sequences, Pacific J. Math. 48 (1973), 197–202
11. L. Olsen, Random geometrically graph directed self-similar multifractals, Longman Scientific and Technical, 1994
12. C. A. Rogers, Hausdorff measures, Cambridge University Press, 1970
13. C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74 https://doi.org/10.1017/S0305004100059119