Note on powers in three interval exchange transformations

We study repetitions in infinite words coding exchange of three intervals with permutation (3, 2, 1), called 3iet words. The language of such words is determined by two parameters, ε,ℓ. We show that finiteness of the index of 3iet words is equivalent to boundedness of the coefficients of the continued fraction of ε. In this case, we also give an upper and a lower estimate on the index of the corresponding 3iet word. Our main tool is the connection between a 3iet word with parameters ε,ℓ and sturmian words with slope ε.