On the trace subshifts of fractional multiplication automata

Abstract

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.