Hausdorff measure of dominated planar self-affine sets with large dimension

Abstract

In this paper, we investigate the Hausdorff measure of planar dominated self-affine sets at their affinity dimension. We show that the Hausdorff measure being positive and finite is equivalent to the Käenmäki measure being a mass distribution. Moreover, under the open bounded neighbourhood condition, we will show that the positivity of the Hausdorff measure is equivalent to the projection of the Käenmäki measure in every Furstenberg direction being absolutely continuous with bounded density. This also implies that the affinity and the Assouad dimension coincide. We will also provide examples for both of the cases when the Hausdorff measure is zero and positive.