Languages invariant under more symmetries: Overlapping factors versus palindromic richness

Abstract

Factor complexity C and palindromic complexity P of infinite words with language closed under reversal are known to be related by the inequality P(n)+P(n+1)≤2+C(n+1)−C(n) for every n∈N. Words for which the equality is attained for every n are usually called rich in palindromes. We show that rich words contain infinitely many overlapping factors. 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 the notion of G-palindromic richness and give several examples of G-rich words, including the Thue–Morse word as well.