Hurwitz correspondences on compactifications of ${\mathcal{M}}_{0,N}$

Hurwitz correspondences are certain multivalued self-maps of the moduli space ${\mathcal{M}}_{0,N}$. They arise in the study of Thurston’s topological characterization of rational functions. We consider the dynamics of Hurwitz correspondences and ask: On which compactifications of ${\mathcal{M}}_{0,N}$ should they be studied? We compare a Hurwitz correspondence $\mathcal{H}$ across various modular compactifications of ${\mathcal{M}}_{0,N}$, and find a weighted stable curves compactification $X_N^{†}$ that is optimal for its dynamics. We use $X_N^{†}$ to show that the $k$th dynamical degree of $\mathcal{H}$ is the absolute value of the dominant eigenvalue of the pushforward induced by $\mathcal{H}$ on a natural quotient of $H_{2k}({\bar{\mathcal{M}}}_{0,N})$.