On the deck groups of iterates of bicritical rational maps

Abstract

Given a rational map $f:\hat{\mathbb{C}}\to \hat{\mathbb{C}}$ on the Riemann sphere, we define $\mathrm{Deck}(f)$ to be the group of Möbius transformations $\mu$ satisfying $f\circ \mu =f$. In this note, we consider the groups $\mathrm{Deck}(f^k)$, where $f$ is a \emph{bicritical} rational map (that is, a rational map with exactly two critical points) and $f^k$ denotes the $k$th iterate of $f$. In particular, we give a complete description of which groups (up to isomorphism) arise as the groups $\mathrm{Deck}(f^k)$ for bicritical rational maps $f$.

arXiv comment16 pages