An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension of the self-affine set is the affinity dimension for Lebesgue almost every choice of translation vectors. Similar statement was proven by Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine measures. I discuss an orthogonal approach, introducing a class of self-affine systems in which, given translation vectors, the same dimension results hold for Lebesgue almost all matrices.
The work is joint with Balazs Barany and Antti Kaenmaki.