Dynamical degrees of Hurwitz correspondences

Abstract

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 Teichmüller 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.

Result strengthened with more applications to complex dynamics, 1 figure, 14 pages

References in corpus (2)

• (Relative) dynamical degrees of rational maps over an algebraic closed field
• Hurwitz correspondences on compactifications of ${\mathcal{M}}_{0,N}$

Cited by in corpus (3)

• The dynamical degree of billiards in an algebraic curve
• Hurwitz correspondences on compactifications of ${\mathcal{M}}_{0,N}$
• Dynamical invariants of monomial correspondences