具有内点的二维自仿射集和唯一性集唯一性集
Two-dimensional self-affine sets with interior points, and the set of uniqueness
摘要：
设$M$是一个$2\times2$实矩阵，其两个特征值的模均小于~1.考虑$\mathbb R^2 \to \mathbb R^2$的两个自仿射压缩映射，\begin{equation*} T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M v + u, \end{equation*}，其中$u\neq0$。我们感兴趣的是由$T_m$和$T_p$生成的迭代函数系统（IFS）的吸引子的性质，即，唯一非空紧集$A$使得$A = T_m(A) \cup T_p(A)$ .我们的两个主要结果如下：1.如果$M$的两个特征值的绝对值都在$2^{-1/4}\approx 0.8409$和$1$之间，且IFS是非退化的，则$A$有非空内部. 2.对于几乎所有的非退化IFS，具有唯一地址的点集具有正的Hausdorff维数-也充分描述了例外情况。本文继续了我们在[11]中开始的工作。
英文摘要：
Let $M$ be a $2\times2$ real matrix with both eigenvalues less than~1 in modulus. Consider two self-affine contraction maps from $\mathbb R^2 \to \mathbb R^2$, \begin{equation*} T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M v + u, \end{equation*} where $u\neq0$. We are interested in the properties of the attractor of the iterated function system (IFS) generated by $T_m$ and $T_p$, i.e., the unique non-empty compact set $A$ such that $A = T_m(A) \cup T_p(A)$. Our two main results are as follows: 1. If both eigenvalues of $M$ are between $2^{-1/4}\approx 0.8409$ and $1$ in absolute value, and the IFS is non-degenerate, then $A$ has non-empty interior. 2. For almost all non-degenerate IFS, the set of points which have a unique address is of positive Hausdorff dimension -- with the exceptional cases fully described as well. This paper continues our work begun in [11].
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。
