Search

Let $(K_1,\ldots,K_n)$ be a Cantor type graph-directed attractors in $\mathbb{R}^d$. By creating an auxiliary graph, we provide an effective criterion of 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 the attractor's geometric approximations. The method is also applicable for more general graph-directed systems. As an application, we are able to determine the connectedness of all $d$-dimensional generalized Sierpi\'nski sponges of which the corresponding IFSs are allowed to have rotational and reflectional components., Comment: 24 pages, 4 figures

In a previous work joint with Dai and Luo, we show that a connected generalized Sierpi\'nski carpet (or shortly a GSC) has cut points if and only if the associated $n$-th Hata graph has a long tail for all $n\geq 2$. In this paper, we extend the above result by showing that it suffices to check a finite number of those graphs to reach a conclusion. This criterion provides a truly "algorithmic" solution to the cut point problem of connected GSCs. We also construct for each $m\geq 1$ a connected GSC with exactly $m$ cut points and demonstrate that when $m\geq 2$, such a GSC must be of the so-called fragile type., Comment: 28 pages, 7 figures

We investigate a homeomorphism problem on a class of self-similar sets called generalized Sierpinski carpets (or shortly GSCs). It follows from two well-known results by Hata and Whyburn that a connected GSC is homeomorphic to the standard Sierpinski carpet if and only if it has no local cut points. On the one hand, we show that to determine whether a given GSC is connected, it suffices to iterate the initial pattern twice. On the other hand, we obtain two criteria: (1) for a connected GSC to have cut points, (2) for a connected GSC with no cut points to have local cut points. With these two criteria, we characterize all GSCs that are homeomorphic to the standard Sierpinski carpet. Our results on cut points and local cut points hold for Baranski carpets, too. Moreover, we extend the connectedness result to Baranski sponges. Thus, we also characterize when a Baranski carpet is homeomorphic to the standard GSC., Comment: 41 pages, 14 figures

In the paper, we provide an effective method for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the contraction vectors is homogeneous., Comment: 12 pages

Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much more challenging and intriguing at the same time. So far the only known results only cover self-similar sets in $\bR$ with no more than 3 branches. In this study we establish results for the Lipschitz equivalence of self-similar sets with touching structures in $\bR$ with arbitrarily many branches. Key to our study is the introduction of a geometric condition for self-similar sets called {\em substitutable}., Comment: 30 pages, 4 figures