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Periodicity, repetitions, and orbits of an automatic sequence

2009; Elsevier BV; Volume: 410; Issue: 30-32 Linguagem: Inglês

10.1016/j.tcs.2009.02.006

1879-2294

Jean‐Paul Allouche, Narad Rampersad, Jeffrey Shallit,

Algorithms and Data Compression

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 squarefree, 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 α, have automatic continued fraction expansions if α does.