论文标题
$ \ mathbb {r}^d $的一类自属瓷砖，即$ d $ - 二维的驯服球
A class of self-affine tiles in $\mathbb{R}^d$ that are $d$-dimensional tame balls
论文作者
论文摘要
我们在$ \ mathbb {r}^d $（$ d \ ge2 $）中研究了一个具有非固定数字的自然植物瓷砖家族，它自然而然地概括了Deng和Lau最初在$ \ Mathbb {r}^2 $中学习的班级，并将其扩展到$ \ \ Mathbb {r} r}^3} $。通过使用Brouwer对域定理的不变性，以及一种我们称为水平距离的工具，我们获得了瓷砖为$ d $ d $维的驯服球的必要条件。这是作者在早期论文中的猜想积极回答的，指出某种类别的自效瓷砖中的成员对$ d $维球的同型在且仅当其内部连接时。
We study a family of self-affine tiles in $\mathbb{R}^d$ ($d\ge2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Deng and Lau in $\mathbb{R}^2$ and its extension to $\mathbb{R}^3}$ by the authors. By using Brouwer's invariance of domain theorem, along with a tool which we call horizontal distance, we obtain necessary and sufficient conditions for the tiles to be $d$-dimensional tame balls. This answers positively the conjecture in an earlier paper by the authors stating that a member in a certain class of self-affine tiles is homeomorphic to a $d$-dimensional ball if and only if its interior is connected.
