arXiv subjects

Shu-qin Zhang

Publications and source records attributed to Shu-qin Zhang.

3 recordsLinked to original sources

arXiv2019

On self-affine tiles whose boundary is a sphere

Let $M$ be a $3\times 3$ integer matrix each of whose eigenvalues is greater than $1$ in modulus and let $\mathcal{D}\subset\mathbb{Z}^3$ be a set with $|\mathcal{D}|=|\det M|$, called digit set. The set equation $MT = T+\mathcal{D}$ uniquely defines a nonempty compact set $T\subset \mathbb{R}^3$. If $T$ has positive Lebesgue measure it is called a $3$-dimensional self-affine tile. In the present paper we study topological properties of $3$-dimensional self-affine tiles with collinear digit set, i.e., with a digit set of the form $\mathcal{D}=\{0,v,2v,\ldots, (|\det M|-1)v\}$ for some $v\in\mathbb{Z}^3\setminus\{0\}$. We prove that the boundary of such a tile $T$ is homeomorphic to a $2$-sphere whenever its set of neighbors in a lattice tiling which is induced by $T$ in a natural way contains $14$ elements. The combinatorics of this lattice tiling is then the same as the one of the bitruncated cubic honeycomb, a body-centered cubic lattice tiling by truncated octahedra. We give a characterization of $3$-dimensional self-affine tiles with collinear digit set having $14$ neighbors in terms of the coefficients of the characteristic polynomial of $M$. In our proofs we use results of R. H. Bing on the topological characterization of spheres.

math.GT↗

arXiv2018

Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule

It is well-known that the constructions of space-filling curves depend on certain substitution rules. For a given self-similar set, finding such rules is somehow mysterious, and it is the main concern of the present paper. Our first idea is to introduce the notion of skeleton for a self-similar set. Then, from a skeleton, we construct several graphs, define edge-to-trail substitution rules, and explore conditions ensuring the rules lead to space-filling curves. Thirdly, we summarize the classical constructions of the space-filling curves into two classes: the traveling-trail class and the positive Euler-tour class. Finally, we propose a general Euler-tour method, using which we show that if a self-similar set satisfies the open set condition and possesses a skeleton, then space-filling curves can be constructed. Especially, all connected self-similar sets of finite type fall into this class. Our study actually provides an algorithm to construct space-filling curves of self-similar sets.

math.GN↗

arXiv2016

Topology of a class of $p2$-crystallographic replication tiles

We study the topological properties of a class of planar crystallographic replication tiles. Let $M\in\mathbb{Z}^{2\times2}$ be an expanding matrix with characteristic polynomial $x^2+Ax+B$ ($A,B\in\mathbb{Z}$, $B\geq 2$) and ${\bf v}\in\mathbb{Z}^2$ such that $({\bf v},M{\bf v})$ are linearly independent. Then the equation $$MT+\frac{B-1}{2}{\bf v} =T\cup(T+{\bf v})\cup (T+2{\bf v})\cup \cdots\cup(T+(B-2){\bf v})\cup(-T) $$ defines a unique nonempty compact set $T$ satisfying $\overline{T^o}=T$. Moreover, $T$ tiles the plane by the crystallographic group $p2$ generated by the $π$-rotation and the translations by integer vectors. It was proved by Leung and Lau in the context of self-affine lattice tiles with collinear digit set that $T\cup (-T)$ is homeomorphic to a closed disk if and only if $2|A|<B+3$. However, this characterization does not hold anymore for $T$ itself. In this paper, we completely characterize the tiles $T$ of this class that are homeomorphic to a closed disk.

math.DS↗