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arXiv · 1403.7659

Profinite automata

Eric Rowland·Reem Yassawi
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Abstract

Many sequences of $p$-adic integers project modulo $p^\alpha$ to $p$-automatic sequences for every $\alpha \geq 0$. 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 $p^\alpha$ 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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BibTeXRIS

Eric Rowland, Reem Yassawi. 2014-03-29. Profinite automata. https://doi.org/10.1016/j.aam.2016.11.005

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