Palindromes in infinite ternary words

Abstract

We study infinite words u over an alphabet $\mathcal{A}$ satisfying the property $\mathcal{P} :~\mathcal{P}(n)+ \mathcal{P}(n+1) = 1+ \#\mathcal{A} {\rm any} n \in \mathbb{N}$ , where $\mathcal{P}(n)$ denotes the number of palindromic factors of length n occurring in the language of u . We study also infinite words satisfying a stronger property $\mathcal{PE}$ : 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 words with 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 $\mathcal{P}$ and moreover 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 ternary alphabet does not imply $\mathcal{PE}$ .