Factor frequencies in generalized Thue-Morse words

We describe factor frequencies of the generalized Thue-Morse word t_{b,m} defined for integers b greater than 1, m greater than 0 as the fixed point starting in 0 of the morphism \phi_{b,m} given by \phi_{b,m}(k)=k(k+1)...(k+b-1), where k = 0,1,..., m-1 and where the letters are expressed modulo m. We use the result of A. Frid, On the frequency of factors in a D0L word, Journal of Automata, Languages and Combinatorics 3 (1998), 29-41 and the study of generalized Thue-Morse words by S. Starosta, Generalized Thue-Morse words and palindromic richness, arXiv:1104.2476v2 [math.CO].Comment: 11 page

Factor frequencies in languages invariant under more symmetries

The number of frequencies of factors of length $n+1$ in a recurrent aperiodic infinite word does not exceed 3\Delta \C(n), where \Delta \C (n) is the first difference of factor complexity, as shown by Boshernitzan. Pelantov\'a together with the author derived a better upper bound for infinite words whose language is closed under reversal. In this paper, we further diminish the upper bound for uniformly recurrent infinite words whose language is invariant under all elements of a finite group of symmetries and we prove the optimality of the obtained upper bound.Comment: 13 page

Palindromes in infinite ternary words

We study infinite words u over an alphabet A satisfying the property P : P(n)+ P(n+1) = 1+ #A for any n in N, where P(n) denotes the number of palindromic factors of length n occurring in the language of u. We study also infinite words satisfying a stronger property PE: every palindrome of u has exactly one palindromic extension in u. For binary words, the properties P and PE coincide and these properties characterize Sturmian words, i.e., words with the complexity C(n)=n+1 for any n in N. In this paper, we focus on ternary infinite words with the language closed under reversal. For such words u, we prove that if C(n)=2n+1 for any n in N, then u satisfies the property P and moreover u is rich in palindromes. Also a sufficient condition for the property PE is given. We construct a word demonstrating that P on a ternary alphabet does not imply PE.Comment: 12 page

On Brlek-Reutenauer conjecture

Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u)=\sum_{n=0}^{\infty} T(n) in which D(u) denotes the defect of u and T(n) denotes C(n+1)-C(n)+2-P(n+1)-P(n), where C and P are the factor and palindromic complexity of u, respectively. Brlek and Reutenauer verified their conjecture for periodic infinite words. We prove the conjecture for uniformly recurrent words. Moreover, we summarize results and some open problems related to defect, which may be useful for the proof of Brlek-Reutenauer Conjecture in full generality