Some Studies On Random Fractal

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.