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. 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.

中文翻译：

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线上通用自适形IFS的Hausdorff测度和Assouad维数

本文考虑在第一级圆柱重叠的实线上的自适形迭代函数系统（IFS）。在自适形 IFS 的空间中，我们通常（在拓扑意义上）表明，如果这样一个系统的吸引子的 Hausdorff 维数小于 1，那么它的适当维数 Hausdorff 测度为零并且其 Assouad 维数等于 1。我们的主要贡献在于表明如果圆柱相交，则 IFS 通常不满足弱分离特性，因此，我们可以应用 Angelevska、Käenmäki 和 Troscheit 的最新结果。这种现象通常适用于横向族（特别是平移族），在自相​​似的情况下，在拓扑和测量理论意义上，在更一般的拓扑意义上的自适形情况下。