随机分形的若干研究
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摘要
博士论文的主要内容由以下四个部分组成:
1.随机自相似集正则性的研究
集合A(?)R~n称为具有正则性如果它的Hausdorff维数与填充维数相同,称为具有强正则性如果它的Hausdorff维数与计盒维数相同。由于具有正则性的集合具有很好的性质,集合的正则性的研究是分形几何的一个重要的研究课题。自相似集的强正则性首先是在开集条件下获得,继而Falconer证明了即使开集条件不满足,自相似集仍然具有强正则性。Berlinkov and Mauldin[1]证明了在开集条件下,随机自相似集具有正则性。丰德军,文志英,吴军[2]证明了一般的Moran集不具有正则性。关于随机自相似集的正则性至今无任何结果。本博士论文的第1章证明了随机自相似集具有强正则性,特别这一结果包含了Falconer的结果。
2.随机Moran集Hausdorff维数的研究
Moran集是一类非常重要的分形集,该类集合的随机化及其研究是80年代以来非常活跃的课题[2,3]。迄今所知的随机Moran集的构造中要求逐阶压缩比独立同分布。但无论从理论还是从应用的角度均应考虑阶压缩比不同分布的情形。我们在第2章引入了满足上述要求的随机Moran集。在关于这种情形的维数的研究,我们需要克服两个本质困难:其一是由于不同分布,以前关于单个鞅的技巧不能使用,为此我们引入了参数鞅并建立了很细致的极限定理;其二是由于逐阶压缩比不同分布导致极限集合也不同分布,从而对证明各阶矩的存在带来很大的困难。为此我们利用参数鞅并引入了随机网测度估计,最后成功地确定了这一类随机Moran集的Hausdorff维数。这一结果同时也将华苏及Marion的有关确定Moran集的结果推广到随机情形。
3.具有混合相依性分枝过程的收敛定理
为研究高分子聚合与渗流,Mandelbrot[4]在70年代引入了一类(?)-相依分支过程,其后由Peyriere[5]与文志英[6,7]系统地研究了它们的性质与应用。与经典的Galton-Watson分支过程相比较有两个差别:其一是过程的状态空间是着色图,其二是相临的个体的生殖不独立,但对于足够远的两个个体的生殖是独立的。由于这些差别,经典的研究Galton-Watson的方法与技巧不再适用。我们在第3章将(?)-相依过程推广到更为一般的情形,它要求同一代任意两个个体的生殖均不独立,即同代个体的生殖具有混合相依
[1] A. Berlinkov and R. D. Mauldin, Packing measure and dimension of random fractals, preprint.
[2] D. J. Feng, Z. Y. Wen and J. Wu, Some dimensional results for homogeneous Moran stes, Science in China, Series A, 43: 5(1997)475-482.
[3] S. Hua, H. Rao, Z. Y. Wen and J. Wu, Dimensions of a Class of Moran Sets, Science of China, Series A, 43: 8 (2000)836-852
[4]. Mandelbrot B., Colliers aleatoires et une alternative aux promenades au hasard sans boucle: les cordommets discrets et fractals, C. R. Acad. Sci. Paris, Ser. A 286, (1978) 933-936.
[5] Peyriere J., Processus de naissance avec interaction des voisins, C. R. Acad. Sci., Paris, t. 289, (1979), 223-224 et 557.
[6] Wen Z. Y., Sur quelques theoremes de convergence du processus de naissance avec interaction des voisins, Bull. Soc. Math. France 114, (1986), 403-429.
[7] Wen, Z. Y., Etude de certains processus de naissance, These de Doctorat, Univ. de Paris-Sud, Orsay, 1986, Publiquation, T86. 2, 1-89.
[8] R Wingren, Concerning a real-valued continuous function on the interval with graph of Hausdorff dimension 2, L'Enseignement Mathematique., 41(1995) 103-110.
This dissertation consists of 4 chapters of which the first three chapters are contributed to the studies of several kinds of random fractals and the fourth chapter is devoted a answer a question posed by Wingren.1. On the Regularity of Random Self-Similar Sets:A set A (?) R~n is called regular if its Hausdorff dimension coincides with its packing dimension, and strong regular if its Hausdorff dimension coincides with its box-counting dimension. Since the sets with regularity have good properties, the study of the regularity becomes an important subject in fractal geometry. It is proved first that the self-similar sets (SSS) have regularity under open set condition (OSC), then Falconer proved further the conclusion holds still even without OSC. Berlincov and Mauldin [1] proved the regularity of random self-similar sets under the OSC. Recently, in 1997, D. J. Feng, Z. Y. Wen and J. Wu [2] proved that there is no regularity for general Moran sets. It remains open whether random Moran sets have regularity even OSC holds. In Chapter 1 of this dissertation, we prove that random self-similar sets are regular in the general case(without OSC), especially, this result generalize Falconer's result to random setting.2. On the Geometric Properties of Random Moran SetsMoran sets is also a kind of crucial fractal sets. There has been a fast growth in general interests in its randomization and its geometric properties since 80's [2, 4]. All the known random Moran sets require that at each step, the contraction vectors have the same distribution. However, it is necessary to consider the case with different distribution at each step, not only from the point of theory but also from view of application. In chapter 2, we introduce the random Moran sets fulfilling the above requirements. There are two essential difficulties in the study of the dimension of this kind of random Moran sets. The first is the associated martingale is not so obvious. This leads us to work more carefully, in fact in our case, in place of a single martingale for random Moran sets, we consider a family of martingales. The second difficulty is the limit random objects also don't have that same distribution, so it's difficult to prove the existence of moments of all order and to give a common bound of these moments.
We also overcome the difficulty by giving the moments estimates of all orders. This estimation plays a key role in the proof of the main results. This result also generalize Hua's work and Marion's work about deterministic Moran sets to the random setting.3. On the Convergence Theorems of Branching Processes with Mixing InteractionsFor studying percolation and polymer, B. Mandelbrot [5] introduced a kind of l-dependent branching process in 70's. Then J. Peyriere [6] and Z. Y. Wen [7, 8] studied their properties and applications systematically. This kind of model has two differences with the classical branching processes: first is that the state space is a colored graph. The second is that dependence between neighbor individuals is permitted unless they are far away enough. Due to these differences, the classical method in studying branching processes is not available. In chapter 3, we introduce a kind of more general branching processes which allow dependence between all individuals in the same generation, that is, the branching processes with mixing interactions. We introduce new method to deal with this kind of processes. We obtain the non-degenerate condition of limiting distribution, the conditions of the existence of moments and a central limit theorem. We determine also the Hausdorff dimension of the corresponding branching sets.4. On the Problem of P. WingrenIn 1995, P.Wingren [9] posed a question that whether there exists a real-valued continuous nowhere differentiate function on [0,1], the graph of which has Hausdorff dimension 1 and locally infinite one-dimensional Hausdorff measure. In chapter 4, we construct a function fulfilling these requirements, therefore answer P.Wingren's question positively.
1.随机自相似集正则性的研究
集合A(?)R~n称为具有正则性如果它的Hausdorff维数与填充维数相同,称为具有强正则性如果它的Hausdorff维数与计盒维数相同。由于具有正则性的集合具有很好的性质,集合的正则性的研究是分形几何的一个重要的研究课题。自相似集的强正则性首先是在开集条件下获得,继而Falconer证明了即使开集条件不满足,自相似集仍然具有强正则性。Berlinkov and Mauldin[1]证明了在开集条件下,随机自相似集具有正则性。丰德军,文志英,吴军[2]证明了一般的Moran集不具有正则性。关于随机自相似集的正则性至今无任何结果。本博士论文的第1章证明了随机自相似集具有强正则性,特别这一结果包含了Falconer的结果。
2.随机Moran集Hausdorff维数的研究
Moran集是一类非常重要的分形集,该类集合的随机化及其研究是80年代以来非常活跃的课题[2,3]。迄今所知的随机Moran集的构造中要求逐阶压缩比独立同分布。但无论从理论还是从应用的角度均应考虑阶压缩比不同分布的情形。我们在第2章引入了满足上述要求的随机Moran集。在关于这种情形的维数的研究,我们需要克服两个本质困难:其一是由于不同分布,以前关于单个鞅的技巧不能使用,为此我们引入了参数鞅并建立了很细致的极限定理;其二是由于逐阶压缩比不同分布导致极限集合也不同分布,从而对证明各阶矩的存在带来很大的困难。为此我们利用参数鞅并引入了随机网测度估计,最后成功地确定了这一类随机Moran集的Hausdorff维数。这一结果同时也将华苏及Marion的有关确定Moran集的结果推广到随机情形。
3.具有混合相依性分枝过程的收敛定理
为研究高分子聚合与渗流,Mandelbrot[4]在70年代引入了一类(?)-相依分支过程,其后由Peyriere[5]与文志英[6,7]系统地研究了它们的性质与应用。与经典的Galton-Watson分支过程相比较有两个差别:其一是过程的状态空间是着色图,其二是相临的个体的生殖不独立,但对于足够远的两个个体的生殖是独立的。由于这些差别,经典的研究Galton-Watson的方法与技巧不再适用。我们在第3章将(?)-相依过程推广到更为一般的情形,它要求同一代任意两个个体的生殖均不独立,即同代个体的生殖具有混合相依
[1] A. Berlinkov and R. D. Mauldin, Packing measure and dimension of random fractals, preprint.
[2] D. J. Feng, Z. Y. Wen and J. Wu, Some dimensional results for homogeneous Moran stes, Science in China, Series A, 43: 5(1997)475-482.
[3] S. Hua, H. Rao, Z. Y. Wen and J. Wu, Dimensions of a Class of Moran Sets, Science of China, Series A, 43: 8 (2000)836-852
[4]. Mandelbrot B., Colliers aleatoires et une alternative aux promenades au hasard sans boucle: les cordommets discrets et fractals, C. R. Acad. Sci. Paris, Ser. A 286, (1978) 933-936.
[5] Peyriere J., Processus de naissance avec interaction des voisins, C. R. Acad. Sci., Paris, t. 289, (1979), 223-224 et 557.
[6] Wen Z. Y., Sur quelques theoremes de convergence du processus de naissance avec interaction des voisins, Bull. Soc. Math. France 114, (1986), 403-429.
[7] Wen, Z. Y., Etude de certains processus de naissance, These de Doctorat, Univ. de Paris-Sud, Orsay, 1986, Publiquation, T86. 2, 1-89.
[8] R Wingren, Concerning a real-valued continuous function on the interval with graph of Hausdorff dimension 2, L'Enseignement Mathematique., 41(1995) 103-110.
This dissertation consists of 4 chapters of which the first three chapters are contributed to the studies of several kinds of random fractals and the fourth chapter is devoted a answer a question posed by Wingren.1. On the Regularity of Random Self-Similar Sets:A set A (?) R~n is called regular if its Hausdorff dimension coincides with its packing dimension, and strong regular if its Hausdorff dimension coincides with its box-counting dimension. Since the sets with regularity have good properties, the study of the regularity becomes an important subject in fractal geometry. It is proved first that the self-similar sets (SSS) have regularity under open set condition (OSC), then Falconer proved further the conclusion holds still even without OSC. Berlincov and Mauldin [1] proved the regularity of random self-similar sets under the OSC. Recently, in 1997, D. J. Feng, Z. Y. Wen and J. Wu [2] proved that there is no regularity for general Moran sets. It remains open whether random Moran sets have regularity even OSC holds. In Chapter 1 of this dissertation, we prove that random self-similar sets are regular in the general case(without OSC), especially, this result generalize Falconer's result to random setting.2. On the Geometric Properties of Random Moran SetsMoran sets is also a kind of crucial fractal sets. There has been a fast growth in general interests in its randomization and its geometric properties since 80's [2, 4]. All the known random Moran sets require that at each step, the contraction vectors have the same distribution. However, it is necessary to consider the case with different distribution at each step, not only from the point of theory but also from view of application. In chapter 2, we introduce the random Moran sets fulfilling the above requirements. There are two essential difficulties in the study of the dimension of this kind of random Moran sets. The first is the associated martingale is not so obvious. This leads us to work more carefully, in fact in our case, in place of a single martingale for random Moran sets, we consider a family of martingales. The second difficulty is the limit random objects also don't have that same distribution, so it's difficult to prove the existence of moments of all order and to give a common bound of these moments.
We also overcome the difficulty by giving the moments estimates of all orders. This estimation plays a key role in the proof of the main results. This result also generalize Hua's work and Marion's work about deterministic Moran sets to the random setting.3. On the Convergence Theorems of Branching Processes with Mixing InteractionsFor studying percolation and polymer, B. Mandelbrot [5] introduced a kind of l-dependent branching process in 70's. Then J. Peyriere [6] and Z. Y. Wen [7, 8] studied their properties and applications systematically. This kind of model has two differences with the classical branching processes: first is that the state space is a colored graph. The second is that dependence between neighbor individuals is permitted unless they are far away enough. Due to these differences, the classical method in studying branching processes is not available. In chapter 3, we introduce a kind of more general branching processes which allow dependence between all individuals in the same generation, that is, the branching processes with mixing interactions. We introduce new method to deal with this kind of processes. We obtain the non-degenerate condition of limiting distribution, the conditions of the existence of moments and a central limit theorem. We determine also the Hausdorff dimension of the corresponding branching sets.4. On the Problem of P. WingrenIn 1995, P.Wingren [9] posed a question that whether there exists a real-valued continuous nowhere differentiate function on [0,1], the graph of which has Hausdorff dimension 1 and locally infinite one-dimensional Hausdorff measure. In chapter 4, we construct a function fulfilling these requirements, therefore answer P.Wingren's question positively.
引文
[1] A. Berlinkov and R. D. Mauldin, Packing measure and dimension of random fractals, preprint.
[2] D. J. Feng, Z. Y. Wen and J. Wu, Some dimensional results for homogeneous Moran stes, Science in China, Series A, 43: 5(1997)475-482.
[3] S. Hua, H. Rao, Z. Y. Wen and J. Wu, Dimensions of a Class of Moran Sets, Science of China, Series A, 43: 8 (2000)836-852
[4]. Mandelbrot B., Colliers aleatoires et une alternative aux promenades au hasard sans boucle: les cordommets discrets et fractals, C. R. Acad. Sci. Paris, Ser. A 286, (1978) 933-936.
[5] Peyriere J., Processus de naissance avec interaction des voisins, C. R. Acad. Sci., Paris, t. 289, (1979), 223-224 et 557.
[6] Wen Z. Y., Sur quelques theoremes de convergence du processus de naissance avec interaction des voisins, Bull. Soc. Math. France 114, (1986), 403-429.
[7] Wen, Z. Y., Etude de certains processus de naissance, These de Doctorat, Univ. de Paris-Sud, Orsay, 1986, Publiquation, T86. 2, 1-89.
[8] R Wingren, Concerning a real-valued continuous function on the interval with graph of Hausdorff dimension 2, L'Enseignement Mathematique., 41(1995) 103-110.
[2] D. J. Feng, Z. Y. Wen and J. Wu, Some dimensional results for homogeneous Moran stes, Science in China, Series A, 43: 5(1997)475-482.
[3] S. Hua, H. Rao, Z. Y. Wen and J. Wu, Dimensions of a Class of Moran Sets, Science of China, Series A, 43: 8 (2000)836-852
[4]. Mandelbrot B., Colliers aleatoires et une alternative aux promenades au hasard sans boucle: les cordommets discrets et fractals, C. R. Acad. Sci. Paris, Ser. A 286, (1978) 933-936.
[5] Peyriere J., Processus de naissance avec interaction des voisins, C. R. Acad. Sci., Paris, t. 289, (1979), 223-224 et 557.
[6] Wen Z. Y., Sur quelques theoremes de convergence du processus de naissance avec interaction des voisins, Bull. Soc. Math. France 114, (1986), 403-429.
[7] Wen, Z. Y., Etude de certains processus de naissance, These de Doctorat, Univ. de Paris-Sud, Orsay, 1986, Publiquation, T86. 2, 1-89.
[8] R Wingren, Concerning a real-valued continuous function on the interval with graph of Hausdorff dimension 2, L'Enseignement Mathematique., 41(1995) 103-110.