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 $\textrm{Deck}(f^k)$ of deck transformations of $f^k$, we show that if $g$ is a bicritical rational map that shares an iterate with $f$, then $\mat...

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Bibliographic Details
Published in:International mathematics research notices 2024-01, Vol.2024 (2), p.1568-1605
Main Authors: Koch, Sarah, Lindsey, Kathryn, Sharland, Thomas
Format: Article
Language:English
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Summary: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 that 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.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnad041