In our earlier paper [Peltomäki and Whiteland (2017) [5]], we introduced a symbolic square root map. Every optimal squareful infinite word s contains exactly six minimal squares and can be written as a product of these squares: $s=X_1^2{X}_2^2\cdots$. The square root $\sqrt{s}$ of s is the infinite word $X_1{X}_2\cdots$ obtained by deleting half of each square. We proved that the square root map preserves the languages of Sturmian words (which are optimal squareful words). The dynamics of the square root map on a Sturmian subshift are well understood. In our earlier work, we introduced another type of subshift of optimal squareful words which together with the square root map form a dynamical system. In this paper, we study these dynamical systems in more detail and compare their properties to the Sturmian case. The main results are characterizations of periodic points and the limit set. The results show that while there is some similarity it is possible for the square root map to exhibit quite different behavior compared to the Sturmian case.

在我们的早期论文[Peltomäkiand Whiteland（2017）[5]]中，我们引入了符号平方根图。每个最佳平方无穷词s恰好包含六个最小平方，可以写成这些平方的乘积：$s=X_{\mathrm{1个}}^2{X}_2^2\cdots$。平方根$\sqrt{s}$的s是无限词$X_{\mathrm{1个}}{X}_2\cdots$通过删除每个正方形的一半获得。我们证明了平方根图保留了Sturmian单词（这是最佳的正方形单词）的语言。充分了解Sturmian子移位的平方根图的动力学。在我们的早期工作中，我们介绍了另一种最优平方词的子移位，它与平方根图一起构成了一个动力系统。在本文中，我们将更详细地研究这些动力学系统，并将其性质与Sturmian情况进行比较。主要结果是表征周期点和极限集。结果表明，尽管存在一些相似性，但平方根图与Sturmian案例相比可能表现出完全不同的行为。