# Cyclic Complexity of Words

Abstract : We introduce and study a complexity function on words $c_x(n),$ called \emph{cyclic complexity}, which counts the number of conjugacy classes of factors of length $n$ of an infinite word $x.$ We extend the well-known Morse-Hedlund theorem to the setting of cyclic complexity by showing that a word is ultimately periodic if and only if it has bounded cyclic complexity. Unlike most complexity functions, cyclic complexity distinguishes between Sturmian words of different slopes. We prove that if $x$ is a Sturmian word and $y$ is a word having the same cyclic complexity of $x,$ then up to renaming letters, $x$ and $y$ have the same set of factors. In particular, $y$ is also Sturmian of slope equal to that of $x.$ Since $c_x(n)=1$ for some $n\geq 1$ implies $x$ is periodic, it is natural to consider the quantity $\liminf_{n\rightarrow \infty} c_x(n).$ We show that if $x$ is a Sturmian word, then $\liminf_{n\rightarrow \infty} c_x(n)=2.$ We prove however that this is not a characterization of Sturmian words by exhibiting a restricted class of Toeplitz words, including the period-doubling word, which also verify this same condition on the limit infimum. In contrast we show that, for the Thue-Morse word $t$, $\liminf_{n\rightarrow \infty} c_t(n)=+\infty.$
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Type de document :
Communication dans un congrès
Csuhaj-Varjú, Ersébet, Dietzfelbinger, Martin, Ésik, Zoltán (Eds.). MFCS 2014 (39th International Symposium), Aug 2014, Budapest, Hungary. ACADEMIC PRESS INC ELSEVIER SCIENCE, 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA, 145, pp.36 - 56, 2017, 〈http://www.inf.u-szeged.hu/mfcs2014/〉. 〈10.1016/j.jcta.2016.07.002〉
Domaine :

https://hal.archives-ouvertes.fr/hal-01024885
Contributeur : Luca Q. Zamboni <>
Soumis le : mercredi 16 juillet 2014 - 18:44:30
Dernière modification le : lundi 13 novembre 2017 - 13:26:01

### Citation

Julien Cassaigne, Gabriele Fici, Marinella Sciortino, Luca Q. Zamboni. Cyclic Complexity of Words. Csuhaj-Varjú, Ersébet, Dietzfelbinger, Martin, Ésik, Zoltán (Eds.). MFCS 2014 (39th International Symposium), Aug 2014, Budapest, Hungary. ACADEMIC PRESS INC ELSEVIER SCIENCE, 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA, 145, pp.36 - 56, 2017, 〈http://www.inf.u-szeged.hu/mfcs2014/〉. 〈10.1016/j.jcta.2016.07.002〉. 〈hal-01024885〉

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