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:
► Refined horoball counting and conformal measure for Kleinian group actions
(with L. Stuart), arXiv
submitted.
► Intermediate dimensions of infinitely generated attractors (with A. Banaji), arXiv
submitted.
► Assouad type dimensions of parabolic Julia sets
(with L. Stuart), arXiv
submitted.
► The Assouad spectrum of Kleinian limit sets and Patterson-Sullivan measure
(with L. Stuart), arXiv
Geometriae Dedicata (to appear).
► A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra (with L. Stuart), arXiv
Bulletin of the American Mathematical Society (to appear).
► Regularity of Kleinian limit sets and Patterson-Sullivan measures, arXiv
Transactions of the American Mathematical Society, 372, (2019), 4977-5009.
► Micromeasure distributions and applications for conformally generated fractals (with M. Pollicott), arXiv
Mathematical Proceedings of the Cambridge Philosophical Society, 159, (2015), 547-566.
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.