EVENT DETAILS AND ABSTRACT

Title: Julia sets having conformal dimension one
Speaker: Insung Park
Speaker Info: ICERM
Brief Description:
Special Note: Pre-seminar will be 3-4 pm and (regular) dynamics seminar will be 4-5 pm
Abstract:

Complex dynamics in one variable is the study of dynamical systems defined by iterations of rational maps on the Riemann sphere. For a critically finite rational map, the Julia set is a fractal defined as the repeller. Being embedded in the sphere, the Julia set of a 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 a variety of Julia sets, including the Julia sets of polynomials and Newton maps. In this talk, we show that a Julia set has conformal dimension one if and only if an invariant graph with topological entropy zero exists. In the spirit of Sullivan’s dictionary, we can also compare this result to the classification of Gromov-hyperbolic groups whose boundaries have conformal dimension one, which is proven by Carrasco-Mackay.

In the pre-seminar, we look at some basic examples of the dynamics of rational maps and their Julia sets. We also discuss important theorems and conjectures, such as Thurston's characterizations, the density of hyperbolicity, and the local connectivity of the Mandelbrot set.