On the Trace Subshifts of Fractional Multiplication Automata

We address the dynamics of the cellular automaton (CA) that multiplies by $p\mathrm{/}q$ 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-$p\mathrm{/}q$ 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.