Istvan Kolossvary

Typical self-similar and self-conformal attractors on the line

In 2001 Peres, Simon and Solomyak considered one-parameter families of self-similar Iterated Function Systems (IFSs) on the line satisfying the so-called transversality condition. They proved that if the similarity dimension is less than 1, then for a typical parameter (both in category and measure sense) the existence of overlaps between cylinders implies that the appropriate dimensional Hausdorff measure of the attractor is zero. We extend this result both for self-similar and self-conformal IFSs on the line. Moreover, combining with recent results of Fraser-Henderson-Olson-Robinson and Angelevska-Kaenmaki-Troscheit we obtain that the Assouad dimension of such systems is 1. Joint work with Balazs Barany, Michal Rams, and Karoly Simon