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

## TL;DR

This paper studies the algebraic stability of certain meromorphic maps on moduli spaces derived from Thurston's pullback maps, linking their dynamics to the original branched coverings on the sphere.

## Contribution

It establishes algebraic stability of these maps on specific Hassett spaces, connecting their dynamics to the properties of the original branched covering maps.

## Key findings

- R_{ extphi} is algebraically stable on the Hassett space for certain configurations.
- The stability relates to the length of the cycle containing the fully ramified point.
- The work links dynamics on moduli space to the original sphere map.

## 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 Teichm\"uller 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.