Karoly Simon, BME Budapest
Title:
Appropriate dimensional Hausdorff measure of typical self-similar and self-conformal
attractors on the line (joint result with Balazs Barany, Istvan Kolossvary and Michal Rams)
Abstract:
In 2001 Peres, Simon and Solomyak considered one-parameter families of self-similar Iterated Function Systems (IFS) on the line satisfying the so-called transversality condition. They proved that if the similarity dimension is smaller than 1, then for a typical parameter (typical both in category and measure sense) the existence of overlaps between the cylinders implies that the appropriate dimensional Hausdorff measure of the attractor is zero. We extend this result both for self-similar and for the self-conformal IFS on the line. Combining these with some very recent theorems of Angelevska, Kaenmaki, and Troscheit we obtain results about the Assouad dimension of the attractors of self-conformal IFS on the line.