Complexity of the language of turns of two arcs. Short communications article

The paper is devoted to the study of the number of words of length n generated by rotations of all possible two arcs of a circle by a fixed irrational angle of rotation 𝜀. In [1], (see also [2], a cubic estimate for the arithmetic complexity of Sturm’s words is obtained, which implies a cubic estimate for the number of words generated by rotations of two arcs. In this paper, the angle of rotation is assumed to be fixed, as a result of which the estimate for the number of words of length n is quadratic in n. This work was is carried out with the help of the Russian Science Foundation Grant N 17-11-01377.