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

TL;DR

This paper studies Hurwitz correspondences on the moduli space of genus zero curves, identifying an optimal compactification for analyzing their dynamics and relating their dynamical degrees to eigenvalues of induced maps.

Contribution

It introduces a weighted stable curves compactification $X_N^$ that optimally supports the dynamics of Hurwitz correspondences and links their dynamical degrees to eigenvalues of induced homology maps.

Findings

Identifies $X_N^$ as the optimal compactification for Hurwitz correspondence dynamics.

Shows the $k$th dynamical degree equals the absolute value of the dominant eigenvalue of a homology map.

Provides a method to compute dynamical degrees via eigenvalues of pushforward maps.

Abstract

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 $X_N^\dagger$ that is optimal for its dynamics. We use $X_N^\dagger$ 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 $H_{2k}(\overline{\mathcal{M}}_{0,N})$.