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 given by a pressure functional 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 translations are fixed and matrices vary, proving that the dimension results hold for Lebesgue almost all matrices.
The work is joint with Balazs Barany and Antti Käenmäki.