Holomorphic dynamics studies the iteration of rational, polynomial, or general entire functions. The Julia set of such a function can informally be though of as the set of all points where the sequence determined by the function and its iterates fails to be equicontinuous, so that nearby points follow different trajectories under iteration. Computer images suggest the Julia set has a rich fractal structure.
In this talk, we will define various notions of dimension (Hausdorff, Minkowski, and packing) used to study fractals. We will discuss relevant dimension results for Julia sets of polynomial/rational functions, and compare these results to what is known about the iteration of (transcendental) non-polynomial entire functions. We will conclude by a discussion of my recent result constructing the first known examples of transcendental entire functions with fractional packing dimension.