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.
|State||Published - 2020|