We continue the exploration of the relationship between conformal dimension and the separation profile by computing the separation of families of spheres in hyperbolic graphs whose boundaries are standard Sierpiński carpets and Menger sponges. In all cases, we show that the separation of these spheres is nd−1 d for some d which is strictly smaller than the conformal dimension, in contrast to the case of rank 1 symmetric spaces of dimension ≥ 3. The value of d obtained naturally corresponds to a previously known lower bound on the conformal dimension of the associated fractal.

中文翻译：

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分形图的分离剖面

我们通过计算边界为标准谢尔宾斯基地毯和门格尔海绵的双曲线图中球体族的分离，继续探索保形尺寸和分离轮廓之间的关系。在所有情况下，我们表明这些球体的分离是nd-1 d对于一些d它严格小于共形维度，与维度为 1 的对称空间的情况相反 ≥ 3. 的价值d获得自然对应于先前已知的相关分形的保形维数的下限。