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

In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a by-product, we give a new proof of the theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved in [A. Carpi, A. de Luca, Special factors, periodicity, and an application to Sturmian words, Acta Inform. 36 (2000) 983–1006] and [D. Damanik, D. Lenz, The index of Sturmian sequences, European J. Combin. 23 (2002) 23–29].