Self-similar sets one are of the most fundamental classes of fractal. They are made up of scaled down copies of themselves, see familiar examples below.
When the scaled down pieces are sufficiently separated, the cylinders provide efficient covers of the set and therefore dimension calculations become straightforward counting problems.
However, when there are overlaps the situation is much more complicated. During my PhD I observed that the Assouad dimension could be distinct from the Hausdorff dimension
provided there are overlaps. Later, in collaboration with Henderson, Olson and Robinson, I proved a dichotomy completely describing the Assouad dimension of
self-similar sets in the line. These papers are here:
► On the Assouad dimension of self-similar sets with overlaps (with A. M. Henderson, E. J. Olson & J. C. Robinson), arXiv
Advances in Mathematics, 273, (2015), 188-214.
► Assouad type dimensions and homogeneity of fractals, arXiv
Transactions of the American Mathematical Society, 366, (2014), 6687-6733.
Other work on self-similar sets and measures includes:
► On the equality of Hausdorff measure and Hausdorff content (with Á. Farkas), arXiv
Journal of Fractal Geometry, 2, (2015), 403-429.
► First and second moments for self-similar couplings and Wasserstein distances, arXiv
Mathematische Nachrichten, 288, (2015), 2028-2041.
Here are the slides I used for a talk about the Assouad dimension of self-similar sets with overlaps:
On the left is the Sierpinksi Triangle and on the right is the Sierpinski Carpet: two familiar examples of self-similar sets.