Special factors, periodicity, and an application to Sturmian words

Let w be a finite word and n the least non-negative integer such that w has no right special factor of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $n$\end{document} and its right factor of length n is unrepeated. We prove that if all the factors of another word v up to the length n + 1 are also factors of w, thenv itself is a factor ofw. A similar result for ultimately periodic infinite words is established. As a consequence, some ‘uniqueness conditions’ for ultimately periodic words are obtained as well as an upper bound for the rational exponents of the factors of uniformly recurrent non-periodic infinite words. A general formula is derived for the ‘critical exponent’ of a power-free Sturmian word. In particular, we effectively compute the ‘critical exponent’ of any Sturmian sequence whose slope has a periodic development in a continued fraction.