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Palindromes in infinite ternary words

Published online by Cambridge University Press:  15 September 2009

L'ubomíra Balková
Affiliation:
Doppler Institute & Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, Praha 2 120 00, Czech Republic; [email protected]
Edita Pelantová
Affiliation:
Doppler Institute & Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, Praha 2 120 00, Czech Republic; [email protected]
Štěpán Starosta
Affiliation:
Institut de Mathématiques de Luminy, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France. Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, Praha 2 120 00, Czech Republic; [email protected]
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Abstract

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

Type
Research Article
Copyright
© EDP Sciences, 2009

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