论文标题
Hausdorff的量度和Assouad尺寸在生产线上的通用自称为IF
Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line
论文作者
论文摘要
本文考虑了第一级圆柱体重叠的真实线上的自符号迭代函数系统（IFSS）。在自我符合条件的IFSS的空间中，我们表明，如果这种系统的吸引子的Hausdorff尺寸小于$ 1 $，则一般（从拓扑意义上）表明，它具有零适当的尺寸Hausdorff量，其补充尺寸等于$ 1 $。我们的主要贡献是表明，如果圆柱体相交，那么IFS通常不满足弱的分离特性，因此我们可以应用Angelevska，Käenmäki和Troscheit的最新结果[BLMS，2020年]。这种现象适用于横向家庭（尤其是翻译家庭），在自相似的情况下，在拓扑和衡量理论意义上，以及在拓扑含义上更一般的自符符号情况下。
This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than $1$ then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to $1$. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit [BLMS, 2020]. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.
