We address the dynamics of the cellular automaton (CA) that multiplies by $p/q$ in base pq (for coprime $p>q>1$) by studying its trace subshift. We present a conjugacy of the trace to a previously studied base-$p/q$ numeration system. We also show that the trace subshift is not synchronizing and in particular not sofic. As a byproduct we compute its complexity function and we conclude by presenting an example of a sofic shift with the same complexity function.

中文翻译：

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关于分数乘法自动机的迹线子移位

我们解决了细胞自动机（CA）乘以 $p/q$在基本pq中（用于互质$p>q>\mathrm{1个}$），通过研究其跟踪子位移。我们将迹线结合到先前研究的基础上，$p/q$计数系统。我们还表明，跟踪子移位未同步，尤其是未同步。作为副产品，我们计算其复杂度函数，最后给出一个具有相同复杂度函数的sofic移动示例。