Hurwitz correspondences on compactifications of M0,N

Hurwitz correspondences are certain multi-valued self-maps of the moduli space M0,N. They arise in the study of Thurston's topological characterization of rational functions. We compare the dynamics of Hurwitz correspondence H on two different compactifications of M0,N: the Deligne-Mumford compactification M‾0,N, as well as a Hassett space of weighted stable curves. We use this comparison to show that the k-th dynamical degree of H is the absolute value of the dominant eigenvalue of the pushforward induced by H on a natural quotient of H2k(M‾0,N).