Thurston obstructions and tropical geometry
Rohini Ramadas
TL;DR
This work builds a bridge between complex dynamics and tropical geometry by formulating a tropical analogue of Thurston’s theory for PCF rational maps. It introduces a tropical Teichmüller space as the cone over the boundary complex of the classical Teichmüller/Moduli framework and defines a tropical pullback map that mirrors the holomorphic Thurston pullback on weighted multicurves. The paper shows that Thurston obstructions correspond to rays fixed (and scaled) by the tropical moduli space correspondence, tying obstructions to tropical eigendata and Hurwitz-equivalent maps. It then situates this within the Hurwitz space framework, constructing tropical Hurwitz spaces and clarifying when tropical and classical dynamics align, particularly in degree two, and discusses extensions to augmented/true tropical settings. Overall, the results provide a conceptual, computationally tractable lens to study the dynamics near infinity in moduli spaces and offer a concrete tropical framework for obstructions and rigidity phenomena in PCF dynamics.
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 Teichmüller 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.