Bicritical rational maps with a common iterate

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 $\text{Deck}(f^k)$ 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.

29 pages, 6 figures