The properties characterizing Sturmian words are considered for words on multiliteral alphabets. We summarize various generalizations of Sturmian words to multiliteral alphabets and enlarge the list of known relationships among these generalizations. We provide a new equivalent definition of rich words and make use of it in the study of generalizations of Sturmian words based on palindromes. We also collect many examples of infinite words to illustrate differences in the generalized definitions of Sturmian words.

Among the various ways to construct a characteristic Sturmian word, one of the most used consists in defining an infinite sequence of prefixes that are standard. Nevertheless in any characteristic word c, some standard words occur that are not prefixes of c. We characterize all standard words occurring in any characteristic word (and so in any Sturmian word) using firstly morphisms, then standard prefixes and finally palindromes.

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 $\mathcal{PE}$ .

We study the palindromic complexity of infinite words uβ, the fixed points of the substitution over a binary alphabet, φ(0) = 0a1, φ(1) = 0b1, with a - 1 ≥ b ≥ 1, which are canonically associated with quadratic non-simple Parry numbers β.

In this paper, we solve some open problems related to (pseudo)palindrome closure operators and to the infinite words generated by their iteration, that is, standard episturmian and pseudostandard words. We show that if ϑ is an involutory antimorphism of A*, then the right and left ϑ-palindromic closures of any factor of a ϑ-standard word are also factors of some ϑ-standard word. We also introduce the class of pseudostandard words with “seed”, obtained by iterated pseudopalindrome closure starting from a nonempty word. We show that pseudostandard words with seed are morphic images of standard episturmian words. Moreover, we prove that for any given pseudostandard word s with seed, all sufficiently long left special factors of s are prefixes of it.