Coding rotations on intervals

We show that the coding of rotation by $\alpha$ on $m$ intervals with rationally independent lengths can be recoded over $m$ Sturmian words of angle $\alpha \mathrm{.}$ More precisely, for a given $m$ an universal automaton is constructed such that the edge indexed by the vector of values of the $i$th letter on each Sturmian word gives the value of the $i$th letter of the coding of rotation.