标题： 控制平面自仿集的 Hausdorff 测度具有较大的维度 显示英文标题

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

摘要： 本文中，我们研究了在仿射维数下的平面控制自仿集的Hausdorff测度。 我们证明了Hausdorff测度为正且有限等价于Käenmäki测度为质量分布。 此外，在开有界邻域条件下，我们将证明Hausdorff测度为正等价于Käenmäki测度在每一个Furstenberg方向上的投影具有绝对连续性，并且具有有界的密度。 这也意味着仿射维数和Assouad维数相等。 我们还将提供当Hausdorff测度为零和正时的例子。 显示英文摘要

摘要： 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 K\"aenm\"aki 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 K\"aenm\"aki 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.