Leírás
In this talk, we investigate the Hausdorff measure of planar self-affine sets at their affinity dimension under the strong separation condition and domination. When the dimension is less than or equal to 1, we show that the positivity of the measure is equivalent to Ahlfors regularity and to a condition called the "projective open set condition". The situation is strictly different when the dimension is strictly greater than 1. In this case, we show that the Hausdorff measure being positive and finite is equivalent to the Käenmäki measure being a mass distribution, and to the projection of the Käenmäki measure in every Furstenberg direction being absolutely continuous with bounded density. We will also provide examples for both cases, when the Hausdorff measure is zero and positive. The talk is based on two papers, one of which is a joint work with Antti Käenmäki and Han Yu.