Substitution shifts generated by $p$-adic integer sequences

Eric Rowland, Reem Yassawi

Published 2014-03-29, updated 2015-03-08Version 2

We set the stage for studying some substitution shifts defined on an infinite alphabet. We consider sequences of $p$-adic integers that project modulo $p^\alpha$ to a $p$-automatic sequence for every $\alpha \geq 0$. Examples include algebraic sequences of integers, which satisfy this property for any prime $p$, and some cocycle sequences, which we show satisfy this property for a fixed $p$. By considering the shift-orbit closure of such a sequence in $\mathbb{Z}_p^\mathbb{N}$, we describe how this shift is a letter-to-letter coding of a shift generated by a constant-length substitution defined on an uncountable alphabet.