并矢Cantor集及其动力学和随机对应集
Dyadic Cantor set and its kinetic and stochastic counterpart
摘要：
首先，我们提出并研究了一个并矢康托集（DCS）和它的动力学对应物，其中一个生成元将一个区间分成两个相等的部分，并以概率$(1-p)$删除一个。然后，在DCS的情况下，生成器在每个步骤应用于所有现有的间隔，并且在动力学DCS的情况下，仅应用于根据间隔大小以概率选择的一个间隔。其次，我们提出了一个随机DCS中，不同的动力学DCS，发电机随机划分的间隔，而不是相等地分为两部分。最后，对模型进行了解析求解，给出了每种情况下分形维数的精确表达式，并指出了分形维数与相应守恒量之间的关系。此外，我们表明，区间大小分布函数在这两个变种的DCS表现出动态标度，我们验证了它的数值使用的数据崩溃的想法。
英文摘要：
Firstly, we propose and investigate a dyadic Cantor set (DCS) and its kinetic counterpart where a generator divides an interval into two equal parts and removes one with probability $(1-p)$. The generator is then applied at each step to all the existing intervals in the case of DCS and to only one interval, picked with probability according to interval size, in the case of kinetic DCS. Secondly, we propose a stochastic DCS in which, unlike the kinetic DCS, the generator divides an interval randomly instead of equally into two parts. Finally, the models are solved analytically; an exact expression for fractal dimension in each case is presented and the relationship between fractal dimension and the corresponding conserved quantity is pointed out. Besides, we show that the interval size distribution function in both variants of DCS exhibits dynamic scaling and we verify it numerically using the idea of data-collapse.
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