Balazs Barany: Dimension of planar self-affine sets and measures

Let us consider a finite set of contracting affinity transformations (IFS) on the plane. It is well known that there exists a unique, non-empty compact set (attractor), which is invariant w.r.t. the IFS. If the open set condition holds and the affinity maps are similarities then Hutchinson's theorem tells us the value of Hausdorff and box counting dimension of the attractor. In this talk, we extend this result for general affinity maps. That is, if the IFS satisfies the strong open set condition and the linear parts act totally irreducibly then the Hausdorff and box counting dimension of the attractor is equal to the affinity dimension. Moreover, the Hausdorff dimension of every fully supported self-affine measure is equal to the Lyapunov dimension. This is a joint work with Mike Hochman and Ariel Rapaport.