Zhiyuan Zhang

Dimension of affine iterated function systems on R^d

In a joint work with Jialun Li and Wenyu Pan, we study the dimensions of self-affine sets and measures arising from affine iterated function systems on R^d. Assuming that the linear parts generate a Zariski-dense subgroup, we prove that, under the strong open set condition, the Hausdorff dimension of the attractor equals its affinity dimension, and the Hausdorff dimension of the associated self-affine measure \mu equals its Lyapunov dimension.
These results follow from a projection theorem requiring no separation condition: for each integer n between 1 and d, and for almost every (d − n)-dimensional subspace V with respect to the corresponding Furstenberg measure, the orthogonal projection of \mu onto the orthogonal complement of V has Hausdorff dimension equal to the minimum of n and the Hausdorff dimension of \mu. In fact, this theorem holds under assumptions weaker than Zariski density: simplicity of the Lyapunov spectrum and a mild transversality condition which we call the Oseledets non-concentration property.