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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. Wepresent a conjugacy of the trace to a previously studied base-$p/q$ numerationsystem. We also show that the trace subshift is not synchronizing and inparticular not sofic. As a byproduct we compute its complexity function and weconclude by presenting an example of a sofic shift with the same complexityfunction.

On the Trace Subshifts of Fractional Multiplication Automata