  For a number field K with absolute Galois group G_K, we consider the action
of G_K on the infinite tree of preimages of a point in K under a degree-two
rational function phi, with particular attention to the case when phi commutes
with a non-trivial Mobius transfomation. In a sense this is a dynamical systems
analogue to the l-adic Galois representation attached to an elliptic curve,
with particular attention to the CM case. Using a result about the
discriminants of numerators of iterates of phi, we give a criterion for the
image of the action to be as large as possible. This criterion is in terms of
the arithmetic of the forward orbits of the two critical points of phi. In the
case where phi commutes with a non-trivial Mobius transfomation, there is in
effect only one critical orbit, and we give a modified version of our
maximality criterion. We prove a Serre-type finite-index result in many cases
of this latter setting.
