Dynamical degrees of Hurwitz correspondences

Let $\phi$ be a post-critically finite branched covering of a two-sphere. By work of Koch, the Thurston pullback map induced by $\phi$ on Teichm\"{u}ller space descends to a multi-valued self-map --- a Hurwitz correspondence $\mathcal{H}_{\phi}$ --- 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 $\mathcal{H}_{\phi}$ is always non-increasing, and the behavior of this sequence is constrained by the behavior of $\phi$ at and near points of its post-critical set.