On the algebraic relations between Mahler functions

In the last years, a number of authors have studied the algebraic relations between the generating series of automatic sequences. It turns out that these series are solutions of Mahler type equations. This paper is mainly concerned with the difference Galois groups of Mahler type equations (these groups reflect the algebraic relations between the solutions of the equations). In particular, we study in detail the equations of order 22 and compute the difference Galois groups of classical equations related to the Baum-Sweet and to the Rudin-Shapiro automatic sequences.