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

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 $X_N^{†}$ 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 $H_{2k}({\overline{\mathcal{M}}}_{0,N})$.

30 pages, major revisions - result strengthened, new introduction, submitted

References in corpus (3)

• (Relative) dynamical degrees of rational maps over an algebraic closed field
• Dynamical degrees of Hurwitz correspondences
• Birational contractions of ${\overline{\mathrm{M}}}_{0,n}$ and combinatorics of extremal assignments

Cited by in corpus (1)

• Dynamical degrees of Hurwitz correspondences