On powers of words occurring in binary codings of rotations

Abstract

We discuss combinatorial properties of a class of binary sequences generalizing Sturmian sequences and obtained as a coding of an irrational rotation on the circle with respect to a partition in two intervals. We give a characterization of those having a finite index in terms of a two-dimensional continued fraction like algorithm, the so-called D -expansion. Then, we discuss powers occurring at the beginning of these words and we prove, contrary to the Sturmian case, the existence of such sequences without any non-trivial asymptotic initial repetition. We also show that any characteristic sequence (that is, obtained as the coding of the orbit of the origin) has non-trivial repetitions not too far from the beginning and we apply this property to obtain the transcendence of the continued fractions whose partial quotients arises from such sequences when the two letters are replaced by distinct positive integers.