Thurston obstructions and tropical geometry

Abstract

We describe an application of tropical moduli spaces to complex dynamics. A post-critically finite branched covering $\varphi$ of $S^2$ induces a pullback map on the Teichm\"uller space of complex structures of $S^2$; this descends to an algebraic correspondence on the moduli space of point-configurations of $\mathbb{C}\mathbb{P}^1$. We make a case for studying the action of the tropical moduli space correspondence by making explicit the connections between objects that have come up in one guise in tropical geometry and in another guise in complex dynamics. For example, a Thurston obstruction for $\varphi$ corresponds to a ray that is fixed by the tropical moduli space correspndence, and scaled by a factor $\ge 1$. This article is intended to be accessible to algebraic and tropical geometers as well as to complex dynamicists.