Groups and Dynamics Seminar

Date: February 23, 2022

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Christopher Bishop, SUNY Stony Brook

Title: Dessins and Dynamics

Abstract: After defining harmonic measure on a planar domain, I will discuss "true trees", i.e., trees drawn in the plane so that every edge has equal harmonic measure and so that these measures are symmetric on each edge. True trees on the 2-sphere are a special case in Grothendieck's theory of dessins d'enfant, where a graph on a topological surface induces a conformal structure on that surface. I will recall the connection between dessins, equilateral triangulations and branched coverings (Belyi's theorem). I will also describe some recent applications of these ideas to holomorphic dynamics: approximating continuua by polynomial Julia sets, finding meromorphic functions with prescribed postcritical orbits, constructing new dynamical systems on hyperbolic Riemann surfaces, building wandering domains for entire functions, and estimating the fractal dimensions of transcendental Julia sets. There will be many pictures, but few proofs.