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

Mignosi, F., Restivo, A. (2012). Characteristic Sturmian words are extremal for the Critical Factorization Theorem. THEORETICAL COMPUTER SCIENCE, 454, 199-205 [10.1016/j.tcs.2012.03.012].

### Characteristic Sturmian words are extremal for the Critical Factorization Theorem

#### Abstract

We prove that characteristic Sturmian words are extremal for the Critical Factorization Theorem (CFT) in the following sense. If p x ( 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 p x ( 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 p x is bounded, and in such a case x is periodic, or p x ( 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 f n , n ≥ 5 , has only one critical point. Indeed we determine the exact number of critical points in any finite standard Sturmian word.
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2012
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Mignosi, F., Restivo, A. (2012). Characteristic Sturmian words are extremal for the Critical Factorization Theorem. THEORETICAL COMPUTER SCIENCE, 454, 199-205 [10.1016/j.tcs.2012.03.012].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10447/73702`