Profinite automata

Rowland, Eric and Yassawi, Reem (2017). Profinite automata. Advances in Applied Mathematics, 85 pp. 60–83.

DOI: https://doi.org/10.1016/j.aam.2016.11.005

Abstract

Many sequences of p-adic integers project modulo p^α 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 p^α 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.

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