关于分形立方体交集的若干研究
On the intersection of fractal cubes
AI总结：
本文研究分形 k-立方体及其相对 l-面的交集，通过图导向相似系统将其表示为吸引子，并据此给出交集维数公式、测度有限性、基数条件，以及有限交性质和树突性的判定条件。
摘要：
我们考虑 n 阶分形 k-立方体的交集以及它们各自相对 l-面的交集。本文的主要结果是关于将此类交集表示为图导向相似系统吸引子的定理，该表示基于与这些立方体对应的单位集合以及 l-面对的交集。作为推论，我们证明了交集的维数公式及其测度有限性的条件。另一个推论给出了交集具有给定基数的条件。应用这些技术，我们得到了分形 k-立方体具有有限交性质的条件，以及分形立方体为树突（dendrite）的条件。
英文摘要：
We consider the intersections of fractal k-cubes of order n and intersections of their respective opposite l-faces. The main result of the paper is the theorem on representation of such intersection as the attractor of a graph-directed system of similarities in terms of the sets of units corresponding to these cubes and intersections of pairs of l-faces. As a corollary, we prove dimension formula for the intersection and the condition of finiteness of its measure. Another corollary gives the conditions under which the intersections have the given cardinality. Applying these techniques, we obtain the conditions under which a fractal k-cube has the finite intersection property and the conditions under which the fractal cube is a dendrite.
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