非自治迭代函数系统的维数理论
Dimension theory of Non-Autonomous iterated function systems
AI总结：
本文定义非自治吸引子（推广Moran集与迭代函数系统吸引子），给出维数估计；对非自治仿射集引入临界值$s^*$和$s_A$，证明上盒维数与Hausdorff维数分别受其约束，并在随机平移情形下几乎必然等于$s_A$。
摘要：
在本文中，我们定义了一类新的分形，称为“非自治吸引子”，它们是经典Moran集和迭代函数系统吸引子的推广。简单地说，我们用压缩映射替换相似映射，并移除Moran结构中的分离假设。我们给出了非自治吸引子的维数估计。此外，我们研究了一类非自治吸引子，称为“非自治仿射集或仿射集”，其中压缩被限制为仿射映射。为了研究此类分形的维数理论，我们定义了两个临界值$s^*$和$s_A$，非自治仿射集的上盒维数和Hausdorff维数分别以$s^*$和$s_A$为上界。与自仿射分形中$s^*=s_A$不同，我们总有$s^*\geq s_A$，且不等式可能严格成立。在某些条件下，我们得到非自治仿射集的上盒维数和Hausdorff维数可能分别等于$s^*$和$s_A$。特别地，我们研究了具有随机平移的非自治仿射集，此类集合的Hausdorff维数几乎必然等于$s_A$。
英文摘要：
In the paper, we define a class of new fractals named ``non-autonomous attractors", which are the generalization of classic Moran sets and attractors of iterated function systems. Simply to say, we replace the similarity mappings by contractive mappings and remove the separation assumption in Moran structure. We give the dimension estimate for non-autonomous attractors. Furthermore, we study a class of non-autonomous attractors, named `` non-autonomous affine sets or affine sets'', where the contractions are restricted to affine mappings. To study the dimension theory of such fractals, we define two critical values $s^*$ and $s_A$, and the upper box-counting dimensions and Hausdorff dimensions of non-autonomous affine sets are bounded above by $s^*$ and $s_A$, respectively. Unlike self-affine fractals where $s^*=s_A$, we always have that $s^*\geq s_A$, and the inequality may strictly hold. Under certain conditions, we obtain that the upper box-counting dimensions and Hausdorff dimensions of non-autonomous affine sets may equal to $s^*$ and $s_A$, respectively. In particular, we study non-autonomous affine sets with random translations, and the Hausdorff dimensions of such sets equal to $s_A$ almost surely.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
