论文标题
关于精确重叠和开放状态的密集混合
On dense intermingling of exact overlaps and the open set condition
论文作者
论文摘要
我们证明，$ \ mathbb {r}^d $中的某些同质仿期迭代功能系统的某些家族具有开放式条件的属性，并且确切的重叠的存在都在翻译参数的空间中很密集。这些示例表明，在Falconer和Jordan-Pollicott-Simon的定理中，在自我效果集和度量的几乎确定的尺寸上，一组非凡的翻译参数可以是密集的集合。该证明结合了有关$ \ mathbb {r}^d $的自我诉讼的文献的结果，以及对某些伯努利卷积的奇异性的经典论证的改编。由于肯尼恩（Kenyon）作为一种特殊情况，我们的结果包括一个一维的例子。
We prove that certain families of homogenous affine iterated function systems in $\mathbb{R}^d$ have the property that the open set condition and the existence of exact overlaps both occur densely in the space of translation parameters. These examples demonstrate that in the theorems of Falconer and Jordan-Pollicott-Simon on the almost sure dimensions of self-affine sets and measures, the set of exceptional translation parameters can be a dense set. The proof combines results from the literature on self-affine tilings of $\mathbb{R}^d$ with an adaptation of a classic argument of Erdős on the singularity of certain Bernoulli convolutions. Our result encompasses a one-dimensional example due to Kenyon which arises as a special case.
