Palindromes in infinite ternary words

Balková, L'ubomíra; Pelantová, Edita; Starosta, Štěpán

doi:10.1051/ita/2009016

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Palindromes in infinite ternary words

Balková, L'ubomíra ; Pelantová, Edita ; Starosta, Štěpán

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 4, pp. 687-702.

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Résumé

We study infinite words $𝐮$ over an alphabet $𝒜$ satisfying the property $𝒫 : 𝒫 (n) + 𝒫 (n + 1) = 1 + # 𝒜 for any n \in ℕ$ , where $𝒫 (n)$ denotes the number of palindromic factors of length $n$ occurring in the language of $𝐮$ . We study also infinite words satisfying a stronger property $𝒫ℰ : every palindrome of 𝐮 has exactly one palindromic extension in 𝐮 .$ For binary words, the properties $𝒫$ and $𝒫ℰ$ coincide and these properties characterize sturmian words, i.e., words with the complexity $𝒞 (n) = n + 1$ for any $n \in ℕ$ . In this paper, we focus on ternary infinite words with the language closed under reversal. For such words $𝐮$ , we prove that if $𝒞 (n) = 2 n + 1$ for any $n \in ℕ$ , then $𝐮$ satisfies the property $𝒫$ and moreover $𝐮$ is rich in palindromes. Also a sufficient condition for the property $𝒫ℰ$ is given. We construct a word demonstrating that $𝒫$ on a ternary alphabet does not imply $𝒫ℰ$ .

EuDML MR Zbl

DOI : 10.1051/ita/2009016

Classification : 68R15
Mots-clés : ternary infinite words, palindromes, generalized sturmian words, rich words

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     author = {Balkov\'a, L'ubom{\'\i}ra and Pelantov\'a, Edita and Starosta, \v{S}t\v{e}p\'an},
     title = {Palindromes in infinite ternary words},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {687--702},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {4},
     year = {2009},
     doi = {10.1051/ita/2009016},
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     zbl = {1191.68476},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1051/ita/2009016/}
}

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Balková, L'ubomíra; Pelantová, Edita; Starosta, Štěpán. Palindromes in infinite ternary words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 4, pp. 687-702. doi : 10.1051/ita/2009016. https://geodesic-test.mathdoc.fr/articles/10.1051/ita/2009016/

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