论文标题
随机的Menger Sponge的预测
Projections of the random Menger sponge
论文作者
论文摘要
从确定性的Menger Sponge开始，使用与产生分形渗滤集的一个随机过程相似，我们得到了随机的Menger Sponge。我们从Hausdorff维度，Lebesgue度量和内部点的存在来检查其正交投影。我们将这些结果作为我们的定理的特殊情况，用于随机自相似IFS。 这些由类似于分形渗透的随机过程遵守，该过程适用于确定性自相似IF的圆柱体，如Arxiv：1212.1345。在本文中，该行上关联的确定性IF是特殊形式的$ \ Mathcal {s} = \ left \ {\ frac {\ frac {1} {l} X+t_i \ right \ right \} _ {i = 1}^{m}^{m}^{m}^{m} $ t_i \ in \ mathbb {q} $。
Using a similar random process to the one which yields the fractal percolation sets, starting from the deterministic Menger sponge we get the random Menger sponge. We examine its orthogonal projections from the point of Hausdorff dimension, Lebesgue measure and existence of interior points. We obtain these results as special cases of our theorems stated for random self-similar IFSs. These are obatained by a random process similar to the fractal percolation, applied for the cylinder sets of a deterministic self-similar IFS, as in arXiv:1212.1345. In this paper the associated deterministic IFS on the line is of the special form $\mathcal{S}=\left\{\frac{1}{L}x+t_i \right\} _{i=1}^{m}$, where $L\in\mathbb{N}$, $L\geq 2$ and $t_i\in\mathbb{Q}$.
