On symmetries of iterates of rational functions

TL;DR

This paper investigates the symmetries of iterated rational functions, showing that for most functions, the symmetry groups of their iterates are finite and bounded, with special focus on their dynamical properties.

Contribution

It establishes finiteness and bounds for symmetry groups of iterates of rational functions, excluding conjugates of monomials, and explores their infinite unions from a dynamical perspective.

Findings

Groups G(A^{ullet k}) are finite and uniformly bounded for non-monomial A.

Infinite unions of symmetry groups reveal interesting dynamical structures.

Results exclude functions conjugate to z^{ }n, focusing on more general cases.

Abstract

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.