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

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