On the deck groups of iterates of bicritical rational maps

TL;DR

This paper characterizes the deck transformation groups of iterates of bicritical rational maps on the Riemann sphere, providing a complete classification of possible groups up to isomorphism.

Contribution

It offers a complete description of the groups Deck(f^k) for bicritical rational maps, detailing which groups can occur as deck groups of iterates.

Findings

Classified all possible deck groups for bicritical rational map iterates.

Identified conditions under which specific groups arise.

Provided a comprehensive framework for understanding symmetries of iterated bicritical maps.

Abstract

Given a rational map $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ on the Riemann sphere, we define $\mathrm{Deck}(f)$ to be the group of M\"obius 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 $f$.