Languages invariant under more symmetries: overlapping factors versus palindromic richness

Abstract

Factor complexity $\mathcal{C}$ and palindromic complexity $\mathcal{P}$ of infinite words with language closed under reversal are known to be related by the inequality $\mathcal{P}(n)+\mathcal{P}(n+1)\le 2+\mathcal{C}(n+1)-\mathcal{C}(n)$ for any $n\in \mathbb{N}$\,. Word for which the equality is attained for any $n$ is usually called rich in palindromes. In this article we study words whose languages are invariant under a finite group $G$ of symmetries. For such words we prove a stronger version of the above inequality. We introduce notion of $G$-palindromic richness and give several examples of $G$-rich words, including the Thue-Morse sequence as well.

22 pages, 1 figure

References in corpus (3)

• Palindromic richness for languages invariant under more symmetries
• On Brlek-Reutenauer conjecture
• Infinite words rich and almost rich in generalized palindromes

Cited by in corpus (9)

• Palindromic richness for languages invariant under more symmetries
• Complementary symmetric Rote sequences: the critical exponent and the recurrence function
• Palindromic closures using multiple antimorphisms
• The repetition threshold for binary rich words
• Morphic images of episturmian words having finite palindromic defect
• Factor frequencies in languages invariant under more symmetries
• Repetitions in infinite palindrome-rich words
• Factor frequencies in generalized Thue-Morse words
• On Words with the Zero Palindromic Defect