On symmetries of iterates of rational functions

Abstract

Let $A$ be a rational function of degree $n\geq 2$. Let us denote by $ G(A)$ the group of Möbius transformations $σ$ such that $ A\circ σ=ν_σ \circ A$ for some Möbius transformations $ν_σ$, and by $Σ(A)$ and ${\rm Aut}(A)$ the subgroups of $ G(A)$ consisting of $σ$ such that $ A\circ σ= A$ and $ A\circ σ= σ\circ A$, correspondingly. In this paper, we study sequences of the above groups arising from iterating $A$. In particular, we show that if $A$ is not conjugate to $z^{\pm n},$ then the orders of the groups $ G(A^{\circ k})$, $k\geq 2,$ are finite and uniformly bounded in terms of $n$ only. We also prove a number of results about the groups $Σ_{\infty}(A)=\cup_{k=1}^{\infty} Σ(A^{\circ k})$ and ${\rm Aut}_{\infty}(A)=\cup_{k=1}^{\infty} {\rm Aut}(A^{\circ k})$, which are especially interesting from the dynamical perspective.