与非连续共线数集相关的平面自仿射集的连通性
Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets
摘要：
本文主要研究了由整数扩张矩阵$A$与$|\det (A)|=3$和共线数字集${\mathcal{D}}=\{0,1,b\}v$生成的平面自仿射集$T(A,{\mathcal{D}})$的连通性，其中$b>1$和$v\in {\mathbb{R}}^2$使得$\{v, Av\}$是线性无关的.本文讨论了数字$b$的定义域，从而确定了$T(A,{\mathcal{D}})$的连通性.特别地，当我们限制$b$为整数时，得到了一个完整的特征。在一般情况下也得到了$|\det (A)|> 3$的一些结果。
英文摘要：
In the paper, we focus on the connectedness of planar self-affine sets $T(A,{\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\det (A)|=3$ and a collinear digit set ${\mathcal{D}}=\{0,1,b\}v$, where $b>1$ and $v\in {\mathbb{R}}^2$ such that $\{v, Av\}$ is linearly independent. We discuss the domain of the digit $b$ to determine the connectedness of $T(A,{\mathcal{D}})$. Especially, a complete characterization is obtained when we restrict $b$ to be an integer. Some results on the general case of $|\det (A)|> 3$ are obtained as well.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
