Profinite automata

Many sequences of p-adic integers project modulo pα to p-automatic sequences for every α≥0. Examples include algebraic sequences of integers, which satisfy this property for every prime p, and some cocycle sequences, which we show satisfy this property for a fixed p. For such a sequence, we construct a profinite automaton that projects modulo pα to the automaton generating the projected sequence. In general, the profinite automaton has infinitely many states. Additionally, we consider the closure of the orbit, under the shift map, of the p-adic integer sequence, defining a shift dynamical system. 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, and we establish some dynamical properties of these shifts.