Dynamical Systems seminar - Lower bounds on the dimension of some quadratic Julia sets

Dates

Wednesday, October 20, 2021 - 14:00 to 15:00

Speaker: Neil Dobbs (University College Dublin)

Abstract: Within hyperbolic components, the Hausdorff dimension of quadratic Julia sets varies analytically. On the boundary of the Mandelbrot set, on the other hand, it varies discontinuously. Indeed, there is a residual set of parameters (in the boundary) where the dimension is 2, and a full harmonic measure set where it is less than 2. For the quadratic $z^2 -2$, the Julia set is an interval and hence has dimension 1. We shall show how to obtain strong lower bounds in a neighbourhood of this map.