Sturmian words and words with a critical exponent

Let S be a standard Sturmian word that is a fixed point of a non-trivial homomorphism. Associated to the infinite word S is a unique irrational number β with 0 <β <1. We prove that the standard Sturmian word S contains no fractional power with exponent greater than Ω and that for any real number ε > 0 it contains a fractional power with exponent greater than Ω - ε; here Ω is a constant that depends on β. The constant Ω is given explicitly. Using these results we are able to give a short proof of Mignosi's theorem and give an exact evaluation of the maximal power that can occur in a standard Sturmian word.