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.