Dynamical Galois groups are constructed by iterating a rational function over a field \(K\) and looking at the tower of preimages of a fixed point of \(P^1\). A couple of years ago, Andrews and Petsche conjectured that when \(K\) is a number field and the function is a polynomial, such groups can only be abelian in trivial cases. This has only been proven to be true in a handful of cases, e.g. when \(K\) is the field of rationals. In this talk, we will consider the same question when \(K\) is a global function field. I will show how, combining a series of reduction steps that involve the construction of Bottcher coordinates in positive characteristic, it is possible to prove that if the rational function \(f\) has a superattracting cycle the dynamical Galois group of \(f\) with basepoint \(\alpha\) can be abelian only if the pair \((f,\alpha)\) is defined over the constant field of \(K\). This is joint work with P. Ingram and C. Pagano.