Relation between powers of factors and the recurrence function characterizing Sturmian words

In this paper we put into relation the index of an infinite aperiodic word and its recurrence function. With the use of this relation, we then give a new characterization of Sturmian words. As a byproduct, we give a new proof of theorem of Damanik and Lenz describing the index of a Sturmian word in terms of the continued fraction expansion of its slope.