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 beta with 0<beta<1. We prove that the standard Sturmian word S contains no fractional power with exponent greater than Omega and that for any real number epsilon>0 it contains a fractional power with exponent greater than Omega - epsilon; here Omega is a constant that depends on beta. The constant Omega 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. (C) 2000 Elsevier Science B.V. All rights reserved.