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Given a rational map $f$ on the Riemann sphere, there exist two naturally defined groups of symmetries. The first is $\mathrm{Aut}(f)$, made up of M\"obius transformations commuting with $f$. The second is $\mathrm{Deck}(f)$, made up of M\"obius transformations which preserve the fibers of $f$. In this talk, we will consider degree $d$ bicritical rational maps. Given such a map, we will ask: when do there exist non-trivial solutions $g$ to the functional equation $f^{\circ n} = g^{\circ n}$? It turns out the answer is closely related to the space $\Sigma_d$ (called the symmetry locus) of bicritical rational maps $f$ for which $\mathrm{Aut}(f)$ is non-trivial. Furthermore, the study makes use of the groups $\mathrm{Deck}(f)$ for the map $f$. The answer to the aforementioned question allows us to make a detailed study of the spaces $\Sigma_d$, giving progress towards answering a question of John Hubbard.

I will show plenty of examples of the phenomena in question, and give a detailed background on holomorphic dynamics for non-experts. This is joint work with Sarah Koch (University of Michigan) and Kathryn Lindsey (Boston College).