Characteristic Sturmian words are extremal for the Critical Factorization Theorem

We prove that characteristic Sturmian words are extremal for the Critical Factorization Theorem (CFT) in the following sense. If denotes the local period of an infinite word x at point n, we prove that x is a characteristic Sturmian word if and only if is smaller than or equal to n+1 for all n≥1 and it is equal to n+1 for infinitely many integers n.This result is extremal with respect to the CFT since a consequence of the CFT is that, for any infinite recurrent word x, either the function is bounded, and in such a case x is periodic, or for infinitely many integers n.As a byproduct of the techniques used in the paper we extend a result of Harju and Nowotka (2002) in [18] stating that any finite Fibonacci word , has only one critical point. Indeed we determine the exact number of critical points in any finite standard Sturmian word.