Regularity of the affinity dimension for some planar iterated function systems

The affinity dimension, introduced by Falconer in the 1980s, is the `typical' value of the Hausdorff dimension of a self-affine set. In 2014, Feng and Shmerkin proved that the affinity dimension is continuous as a function of the maps defining the self-affine set, thus resolving a long-standing open problem in the fractal geometry community. In this talk we will discuss stronger regularity properties of the affinity dimension in some special cases. This is based on recent work with Ian Morris.