Self-affine sets and measures

Self-affine sets are attractors of IFSs consisting of affine contractions. They are a well-studied and important class of fractal and are notoriously difficult to study since affine maps can distort by different amounts in different directions. This is a particular case of the more general class of non-conformal attractors. I have worked a lot with self-affine sets and associated self-affine measures. I am especially interested in planar carpet type sets, such as those introduced by Bedford and McMullen. I wrote a survey article on Bedford-McMullen carpets, which contains some open problems: Some work on box and Hausdorff dimensions of self-affine sets includes: Some work on the Assouad type dimensions of (random and deterministic) self-affine sets includes: Some work on self-affine measures, especially the Lq spectra and Hausdorff dimension: Here are some slides from talks I've given on self-affine sets and measures:

On the left is a self-affine carpet of the type introduced by Bedford and McMullen and on the right is another planar self-affine set.