On symmetries of iterates of rational functions

Fedor Pakovich

Published 2020-06-15Version 1

Let $A$ be a rational function of degree $n\geq 2$. We denote by $ G(A)$ the group of M\"obius transformations $\sigma$ such that $ A\circ \sigma=\nu \circ A$ for some M\"obius transformations $\nu$, and by $\Sigma(A)$ and ${\rm Aut}(A)$ subgroups of $ G(A)$, consisting of M\"obius transformations $\sigma$ such that $ A\circ \sigma= A$ and $ A\circ \sigma= \sigma \circ A$, correspondingly. We show that, unless $A$ has a very special form, the orders of the groups $ G(A^{\circ k})$, $k\geq 1,$ 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 $\Sigma_{\infty}(A)=\cup_{k=1}^{\infty} \Sigma(A^{\circ k})$ and ${\rm Aut}_{\infty}(A)=\cup_{k=1}^{\infty} {\rm Aut}(A^{\circ k})$, 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.