The Thue-Morse Transform

We define the Thue-Morse transform T on a class of infinite binary words. It sends the alternating word a_0 = 010101… to the Thue-Morse sequence. We then study its orbit a_m = T^m(a_0) as well as the sequences u_m and v_m giving respectively the positions of the ones and the zeros in a_m. We obtain an explicit formula for a_m and deduce Prouhet-Tarry-Escott identities for the partitions induced by u_m and v_m. We also give composition formulas for u_m and v_m, and a full description of the factor complexity of a_m.