Let $A$ be a rational function of degree $n\geq 2$. Let us denote by $ G(A)$ the group of M\"obius transformations $\sigma$ such that $ A\circ \sigma=\nu_{ \sigma} \circ A$ for some M\"obius transformations $\nu_{ \sigma}$, and by $\Sigma(A)$ and ${ \rm Aut}(A)$ the subgroups of $ G(A)$ consisting of $\sigma$ such that $ A\circ \sigma= A$ and $ A\circ \sigma= \sigma \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 $\Sigma_{ \infty}(A)=\cup_{ k=1}^{ \infty} \Sigma(A^{ \circ k})$ and ${ \rm Aut}_{ \infty}(A)=\cup_{ k=1}^{ \infty} { \rm Aut}(A^{ \circ k})$, which are especially interesting from the dynamical perspective.
Categories: math.DS math.CV