Steffen Rohde: Mating of trees in analysis, probability and dynamics

Abstract: "Mating of trees", the construction of spheres from two trees, is hard to visualize. It has played a prominent role in hyperbolic geometry and in complex dynamics long before it became the powerful tool in probability theory that it is today.

Very recent work of Danny Calegari and coauthors provides an elegant framework. I will briefly discuss some of this background, and then describe a concrete and elementary example that actually allows for an easy visualization of the mating of trees phenomenon. Despite its simplicity (it can be conveniently described in terms of an iterated function system), it shares a number of features with both the Duplantier-Miller-Sheffield Peanosphere and the mating constructions from complex dynamics.