Iterated function systems (IFS) are commonly used to produce fractals. While self-similar IFS are well studied, self-affine IFS are still relatively new.
In a recent paper Kevin Hare and I considered a simple family of two-dimensional self-affine sets ($=$ attractors of self-affine IFS) and proved several results on their connectedness, interior points, convex hull and corresponding simultaneous expansions. A great deal of natural questions (simple connectedness, set of uniqueness, dimensions, etc.) remain open - even for this most natural family.
The project is aimed at closing these gaps as well as generalising our results to other 2D families (which are completely classified) as well as higher dimensions.