  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.
