Conformal dynamics

Kleinian groups are discrete groups of isometries acting on hyperbolic space (Fuchsian if the hyperbolic space is 2-dimensional). They act conformally on the boundary at infinity and give rise to beautiful fractal limit sets. Rational maps act conformally on the Riemann sphere (complex plane plus a point at infinity) and give rise to beautiful fractal Julia sets. I have studied fractal properties of both of these objects, as well as the Sullivan dictionary which seeks connection between the two settings. I've also worked on self-conformal sets. Some papers include: I gave a talk about my work with Liam Stuart on the Sullivan dictionary at the Seminaire Cristolien d'Analyse Multifractale, CNRS, France, 17 December 2020. Here is a video and here are the slides.

On the left is the limit set of a Kleinian group and on the right is a Julia set of a parabolic rational map.