Bicritical rational maps with a common iterate

TL;DR

This paper investigates bicritical rational maps, showing that sharing an iterate implies identical critical sets and that maps sharing an iterate are closely related, especially in even degree cases, linking to the symmetry locus.

Contribution

It proves that bicritical rational maps sharing an iterate have identical critical and critical value sets, and that in even degree, sharing an iterate implies sharing a second iterate and belonging to the symmetry locus.

Findings

Sharing an iterate implies identical critical sets and critical values.

In even degree, sharing an iterate implies sharing a second iterate.

Maps sharing an iterate belong to the symmetry locus.

Abstract

Let $f$ be a degree $d$ bicritical rational map with critical point set $\mathcal{C}_f$ and critical value set $\mathcal{V}_f$. Using the group $\textrm{Deck}(f^k)$ of deck transformations of $f^k$, we show that if $g$ is a bicritical rational map which shares an iterate with $f$ then $\mathcal{C}_f = \mathcal{C}_g$ and $\mathcal{V}_f = \mathcal{V}_g$. Using this, we show that if two bicritical rational maps of even degree $d$ share an iterate then they share a second iterate, and both maps belong to the symmetry locus of degree $d$ bicritical rational maps.