Proof of the Brlek-Reutenauer conjecture

Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u)[email protected]?"n"="0^+^~T"u(n) in which D(u) denotes the defect of u and T"u(n) denotes C"u(n+1)-C"u(n)+2-P"u(n+1)-P"u(n), where C"u and P"u are the factor and palindromic complexity of u, respectively. This conjecture was verified for periodic words by Brlek and Reutenauer themselves. Using their results for periodic words, we have recently proved the conjecture for uniformly recurrent words. In the present article we prove the conjecture in its general version by a new method without exploiting the result for periodic words.