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Summary: Periodicity, Repetitions, and Orbits of an
Automatic Sequence
Jean-Paul Allouche a
, Narad Rampersad b
, Jeffrey Shallit c,
aCNRS, LRI, UMR 8623, Universit´e Paris-Sud, B^atiment 490, F-91405 Orsay
Cedex, France
bDepartment of Mathematics and Statistics, University of Winnipeg, 515 Portage
Avenue, Winnipeg, MB R3B 2E9, Canada
cSchool of Computer Science, University of Waterloo, Waterloo, Ontario N2L
3G1, Canada
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
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 , have automatic
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