Moran集的中间维数及其可视化
Intermediate dimensions of Moran sets and their visualization
AI总结：
针对Moran集因缺乏遍历性导致中间维数不一定存在的问题，本文计算其上、下中间维数，给出齐次情形的简化公式，并用Möbius变换实现可视化。
摘要：
中间维数是一类新的分形维数，它提供了一个在Hausdorff维数与盒计数维数之间插值的维数谱。本文研究Moran集的中间维数。Moran集可视为自相似集的推广：其在每一层使用不同类别的相似映射生成，且平移不固定，这导致Moran集缺乏遍历性质。因此，中间维数不一定存在，我们计算了Moran集的上、下中间维数。特别地，我们得到了齐次Moran集的简化中间维数公式。此外，我们研究了某些齐次Moran集的上中间维数的可视化，并证明它们的上中间维数由Möbius变换给出。
英文摘要：
Intermediate dimensions are a class of new fractal dimensions which provide a spectrum of dimensions interpolating between the Hausdorff and box-counting dimensions. In this paper, we study the intermediate dimensions of Moran sets. Moran sets may be regarded as a generalization of self-similar sets generated by using different class of similar mappings at each level with unfixed translations, and this causes the lack of ergodic properties on Moran set. Therefore, the intermediate dimensions do not necessarily exist, and we calculate the upper and lower intermediate dimensions of Moran sets. In particular, we obtain a simplified intermediate dimension formula for homogeneous Moran sets. Moreover, we study the visualization of the upper intermediate dimensions for some homogeneous Moran sets, and we show that their upper intermediate dimensions are given by Mobius transformations.
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。
