Languages invariant under more symmetries: Overlapping factors versus palindromic richness

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.