Henna Koivusalo: Dimension of self-affine sets for fixed translation vectors

In 1988, Falconer proved that, for a fixed collection of matrices, the Hausdorff dimension of the corresponding self-affine set is the affinity dimension for Lebesgue almost every choice of translation vectors.
Similar statement was proved by Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine measures. I discuss an orthogonal approach, introducing a class of affine iterated function systems in which, given translation vectors, for Lebesgue almost all matrices, the dimension of the corresponding self-affine set is the affinity dimension. Our proofs rely on Ledrappier-Young theory that was recently verified for affine iterated function systems by Barany and Kaenmaki, and a new transversality condition. In particular our argument does not directly depend on properties of the Furstenberg measure which allows our results to hold for self-affine sets and measures in any Euclidean space and not just in the plane.
The work is joint with Balazs Barany and Antti Kaenmaki.