Characteristic Sturmian words are extremal for the Critical Factorization Theorem

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We prove that characteristic Sturmian words are extremal for the Critical Factorization Theorem (CFT) in the following sense. If px(n) 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 px(n) 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 px is bounded, and in such a case x is periodic, or px(n)≥n+1 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 fn,n≥5, has only one critical point. Indeed we determine the exact number of critical points in any finite standard Sturmian word.