Proof of Brlek-Reutenauer conjecture

TL;DR

This paper proves the Brlek-Reutenauer conjecture for all infinite words with language closed under reversal, establishing a fundamental relationship between defect and complexity measures using a novel proof method.

Contribution

The paper provides the first proof of the conjecture in its full generality without relying on previous results for periodic or uniformly recurrent words.

Findings

The conjecture holds for all infinite words with language closed under reversal.

A new proof method is introduced that does not depend on periodic word results.

The relationship between defect and complexity is rigorously established.

Abstract

Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u) = \sum_{n=0}^{\infty}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.