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Algebraic stability of meromorphic maps descended from Thurston's pullback maps

Let $ϕ:S^2 \to S^2$ be an orientation-preserving branched covering whose post-critical set has finite cardinality $n$. If $ϕ$ has a fully ramified periodic point $p_{\infty}$ and satisfies certain additional conditions, then, by work of Koch, $ϕ$ induces a meromorphic self-map $R_ϕ$ on the moduli space $\mathcal{M}_{0,n}$; $R_ϕ$ descends from Thurston's pullback map on Teichmüller space. Here, we relate the dynamics of $R_ϕ$ on $\mathcal{M}_{0,n}$ to the dynamics of $ϕ$ on $S^2$. Let $\ell$ be the length of the periodic cycle in which the fully ramified point $p_{\infty}$ lies; we show that $R_ϕ$ is algebraically stable on the heavy-light Hassett space corresponding to $\ell$ heavy marked points and $(n-\ell)$ light points.