Dynamical degrees of Hurwitz correspondences

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

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