收稿日期: 2012-03-01
修回日期: 2012-10-22
网络出版日期: 2013-07-15
基金资助
Supported by the National Natural Science Foundation of China (11171342)
On the open set condition of the graph directed self-conformal fractals
Received date: 2012-03-01
Revised date: 2012-10-22
Online published: 2013-07-15
郑水草 . 关于图有向自保形分形的开集条件[J]. 中国科学院大学学报, 2013 , 30(4) : 443 -449 . DOI: 10.7523/j.issn.2095-6134.2013.04.003
In this paper, some conditions equivalent to the open set condition of the graph directed self-conformal fractals are obtained. Especially, it is proven that the strong open set condition and the open set condition are equivalent for such a fractal.
Key words： self-conformal fractal; open set condition; Hausdorff measure
[1] Mauldin R D, Williams S C. Random recursive constructions: asymptotic geometric and topological properties[J]. Trans Amer Math Soc, 1986, 295: 325-346.
[2] Peres Y, Rams M, Simon K, et al. Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets[J]. Proc Amer Math Soc, 2001, 129: 2689-2699.
[3] Hutchinson J E. Fractals and self-similarity[J]. Indiana Univ Math J, 1981, 30: 713-747.
[4] Wang J L. The open set condition for graph directed self-similar sets[J]. Random and Computational Dynamics, 1997, 5(4): 283-305.
[5] Lalley S P. The packing and covering function of some self-similar fractals[J]. Indiana Univ Math J, 1988, 37(3): 699-709.
[6] Arbeiter M, Patzschke N. Ramdom self-similar multifractals[J]. Math Nachr, 1996, 181: 5-42.
[7] Patzschke N. The strong open set condition for self-conformal random fractals[J]. Proc Amer Math Soc, 2003, 131: 2347-2358.
[8] Bandt C, Graf S. Self-similar sets 7: A characterization of self-similar fractals with positive Hausdorff measure[J]. Proc Amer Math Soc, 1992, 114: 995-1001.
[9] Patzschke N. Self-conformal multifractal measures[J]. Advances in Applied Mathematics, 1997, 19: 486-513.
[10] Bowen R. Equililibrium states and the ergodic theorey of Anosov diffeomorphisms[J]. Lecture Notes in Math, 1975, 470.
[11] Falconer K J. Fractal geometry [M]. New York: John Wiley & Sons, 1990.
[12] Rogers C A, Taylor S J. Functions continuous and singular with respect to a Hausdorff measure[J]. Marhematika, 1961, 8: 1-31.
[13] Zheng S C, Hu D H. A note on the multifractal decomposition of directed graph self-similar sets[J]. Wuhan University Journal: Natural Sciences, 2004, 9: 269-272.
/
| 〈 |
|
〉 |
