Dynamical degrees of Hurwitz correspondences

Let $\varphi$ be a post-critically finite branched covering of a two-sphere. By work of Koch, the Thurston pullback map induced by $\varphi$ on Teichm"{u}ller space descends to a multi-valued self-map — a Hurwitz correspondence ${\mathcal{H}}_{\varphi }$ — of the moduli space ${\mathcal{M}}_{0,P}$. We study the dynamics of Hurwitz correspondences via numerical invariants called dynamical degrees. We show that the sequence of dynamical degrees of ${\mathcal{H}}_{\varphi }$ is always non-increasing, and the behavior of this sequence is constrained by the behavior of $\varphi$ at and near points of its post-critical set.