标题： 加法自动机和对应于扩展有理矩阵的数字系统的吸引子 显示英文标题

标题： Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices

摘要： 设$A$为一个具有有理数元素的扩展$2 \times 2$矩阵，并且$\mathbb{Z}^2[A]$为包含$\mathbb{Z}^2$的最小$A$-不变$\mathbb{Z}$-模。 设$\mathcal{D}$为$\mathbb{Z}^2[A]$的一个有限子集，它是一个$\mathbb{Z}^2[A]/A\mathbb{Z}^2[A]$的完全剩余系。 该对$(A,\mathcal{D})$被称为带有{\em 基础} $A$ 和{\em 数字集合} $\mathcal{D}$ 的{\em 位数系统}。 众所周知，每个向量$x \in \mathbb{Z}^2[A]$都可以唯一地写成形式\[ x = d_0 + Ad_1 + \cdots + A^kd_k + A^{k+1}p, \]，其中$k\in \mathbb{N}$是最小的，$d_0,\dots,d_k \in \mathcal{D}$，以及$p$取自一个有限集的{\em 周期性元素}，即所谓的{\em 吸引子}的$(A,\mathcal{D})$。 如果 $p$ 总是可以被选择为 $0$，我们说 $(A,\mathcal{D})$ 具有 {\em 有限性性质}。 在本文中，我们引入有限状态转换自动机，这些自动机在具有共线数字集的数系统$(A,\mathcal{D})$中实现向量$\pm(1,0)^\top$和$\pm(0,1)^\top$与给定向量$x\in \mathbb{Z}^2[A]$的加法。 这些自动机用于表征所有具有有限性性质的对$(A,\mathcal{D})$，并更一般地用于表征这些数字系统的吸引子。 显示英文摘要

摘要： Let $A$ be an expanding $2 \times 2$ matrix with rational entries and $\mathbb{Z}^2[A]$ be the smallest $A$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^2$. Let $\mathcal{D}$ be a finite subset of $\mathbb{Z}^2[A]$ which is a complete residue system of $\mathbb{Z}^2[A]/A\mathbb{Z}^2[A]$. The pair $(A,\mathcal{D})$ is called a {\em digit system} with {\em base} $A$ and {\em digit set} $\mathcal{D}$. It is well known that every vector $x \in \mathbb{Z}^2[A]$ can be written uniquely in the form \[ x = d_0 + Ad_1 + \cdots + A^kd_k + A^{k+1}p, \] with $k\in \mathbb{N}$ minimal, $d_0,\dots,d_k \in \mathcal{D}$, and $p$ taken from a finite set of {\em periodic elements}, the so-called {\em attractor} of $(A,\mathcal{D})$. If $p$ can always be chosen to be $0$ we say that $(A,\mathcal{D})$ has the {\em finiteness property}. In the present paper we introduce finite-state transducer automata which realize the addition of the vectors $\pm(1,0)^\top$ and $\pm(0,1)^\top$ to a given vector $x\in \mathbb{Z}^2[A]$ in a number system $(A,\mathcal{D})$ with collinear digit set. These automata are applied to characterize all pairs $(A,\mathcal{D})$ that have the finiteness property and, more generally, to characterize the attractors of these digit systems.