论文标题
Sierpiński海绵与旋转和反射组件以及相关图导向系统的连接性
The connectedness of Sierpiński sponges with rotational and reflectional components and associated graph-directed systems
论文作者
论文摘要
我们提供了两种方法来表征所有$ d $ d $二二维的SierpińskiSponges的连接性，其相应的IFS被允许具有旋转和反射组件。我们的方法是将其减少到以图指导吸引子的坐标之间的相交问题。更准确地说，让$（k_1，\ ldots，k_n）$为$ \ mathbb {r}^d $中的cantor型吸引子。通过创建辅助图，我们为每对$ 1 \ leq i，j \ leq n $是否为$ k_i \ cap k_j $提供有效标准。此外，可以通过仅检查吸引子的几何近似值来检查空虚。该方法还适用于更通用的图形定向系统。
We provide two methods to characterize the connectedness of all $d$-dimensional generalized Sierpiński sponges whose corresponding IFSs are allowed to have rotational and reflectional components. Our approach is to reduce it to an intersection problem between the coordinates of graph-directed attractors. More precisely, let $(K_1,\ldots,K_n)$ be a Cantor-type graph-directed attractor in $\mathbb{R}^d$. By creating an auxiliary graph, we provide an effective criterion for whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leq n$. Moreover, the emptiness can be checked by examining only a finite number of geometric approximations of the attractor. The approach is also applicable to more general graph-directed systems.
