About this video

This talk was part of the Dynamical Systems seminar at the University of Liverpool.

https://www.liverpool.ac.uk/mathematical-sciences/research/pure-mathematics/research/dynamical-systems/seminars/

Abstract: As a fractal embedded in the Riemann sphere, the Julia set of a post-critically finite rational map has conformal dimension between 1 and 2. The Julia set has conformal dimension 2 if and only if it is the entire Riemann sphere. However, the other extreme case, when conformal dimension=1, contains diverse Julia sets, including the Julia sets of post-critically finite polynomials and Newton maps. In this talk, we show that for a post-critically finite hyperbolic rational map $f$, the Julia set $J_f$ has conformal dimension one if and only if there exists an f-invariant graph with topological entropy zero. In the spirit of Sullivan’s dictionary, we can also compare this result with the classification of Gromov-hyperbolic groups whose boundaries have conformal dimension one, which Carrasco-Mackay proved.