Advanced Search

We prove that the generalized Thue-Morse word ${𝐭}_{b,m}$ defined for $b\ge 2$ and $m\ge 1$ as ${𝐭}_{b,m}={\left(s_b\left(n\right)modm\right)}_{n=0}^{+\infty }$, where $s_b\left(n\right)$ denotes the sum of digits in the base-$b$ representation of the integer $n$, has its language closed under all elements of a group $D_m$ isomorphic to the dihedral group of order $2m$ consisting of morphisms and antimorphisms. Considering antimorphisms $\Theta \in D_m$, we show that ${𝐭}_{b,m}$ is saturated by $\Theta$-palindromes up to the highest possible level. Using the generalisation of palindromic richness recently introduced by the author and E. Pelantová,...

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...

We study infinite words over an alphabet $𝒜$ satisfying the property $𝒫:𝒫\left(n\right)+𝒫(n+1)=1+\#𝒜\mathrm{for}\mathrm{any}n\in \mathbb{N}$, where $𝒫\left(n\right)$ denotes the number of palindromic factors of length occurring in the language of . We study also infinite words satisfying a stronger property $\mathrm{𝒫ℰ}$: every palindrome of has exactly one palindromic extension in . For binary words, the properties $𝒫$ and $\mathrm{𝒫ℰ}$ coincide and these properties characterize Sturmian words, i.e., words with the complexity + 1 for any $n\in \mathbb{N}$. In this paper, we focus on ternary infinite words with...

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...