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

Abstract:

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 Teichmüller 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.