On symmetries of iterates of rational functions

Let A be a rational function of degree n≥2. We denote by G(A) the group of Möbius transformations σ such that A∘σ=ν∘A for some Möbius transformations ν, and by Σ(A) and Aut(A) subgroups of G(A), consisting of Möbius transformations σ such that A∘σ=A and A∘σ=σ∘A, correspondingly. We show that, unless A has a very special form, the orders of the groups G(A^∘k), k≥1, are finite and uniformly bounded in terms of n only. We also prove a number of results allowing us in some cases to calculate explicitly the groups Σ∞(A)=∪^∞_k=1Σ(A^∘k) and Aut_∞(A)=∪^∞_k=1Aut(A^∘k), especially interesting from the dynamical perspective. In addition, we prove that the number of rational functions B of degree d sharing an iterate with A is finite and bounded in terms of n and d only.