Periodicity, repetitions, and orbits of an automatic sequence

TL;DR

This paper simplifies proofs of key properties of automatic sequences, including periodicity and pattern avoidance, and explores automaticity in related sequences and measures, advancing understanding of their decidability and structural features.

Contribution

It provides a simple proof of the decidability of periodicity in automatic sequences and extends automaticity results to orbit closures and related measures.

Findings

Decidability of ultimate periodicity for k-automatic sequences

Decidability of overlap-freeness and pattern avoidance in automatic sequences

Automaticity of quantities like critical exponent and irrationality measure for Sturmian words

Abstract

We revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given k-automatic sequence is ultimately periodic. We prove that it is decidable whether a given k-automatic sequence is overlap-free (or squareefree, or cubefree, etc.) We prove that the lexicographically least sequence in the orbit closure of a k-automatic sequence is k-automatic, and use this last result to show that several related quantities, such as the critical exponent, irrationality measure, and recurrence quotient for Sturmian words with slope alpha, have automatic continued fraction expansions if alpha does.