Towards Transcendental Thurston theory

In the 1980s, William Thurston proved his celebrated topological characterization of rational maps, later written down by Douady and Hubbard. Roughly speaking, Thurston’s theorem gives a criterion for deciding when a topological dynamical system on the two-dimensional sphere induced by a branched covering is equivalent, in an appropriate dynamical sense, to a holomorphic dynamical system given by a rational map on the Riemann sphere. The theorem applies to postcritically finite maps, meaning that every critical point has a finite forward orbit (i.e., it is pre-periodic). It has had a profound influence on the study of rational dynamics, both through its statement and through the ideas introduced in its proof. A natural question is whether Thurston’s theorem can be extended beyond rational maps to transcendental (non-algebraic) functions, such as entire or meromorphic functions. The transcendental setting is considerably more difficult, and a complete analogue of Thurston’s theorem is still unknown. Nevertheless, substantial progress has been made in recent years. In this talk, I will first introduce the main ideas of Thurston theory and will then discuss recent joint work with L. Rempe, in which we show that an analogue of Thurston’s theorem holds for a broad class of functions known as transcendental Belyi maps.