论文标题
中级$β$ -SHIFTS作为贪婪$β$ - 带孔
Intermediate $β$-shifts as greedy $β$-shifts with a hole
论文作者
论文摘要
我们表明，每个中间$β$转化在拓扑上都是与零孔的贪婪$β$转化的共轭，并提供了反例，说明该通信不是一对一。该表征用于（1）建立一个嵌入中间$β$转换定理的krieger，并补充了李，萨尔斯滕，塞缪尔和斯坦纳[2019]的结果，以及（2）获得新的指标和拓扑成果，对ZEROR和ZAL的lie gorment and a kal a kal的幸存者集合，KAL，KAL，KAL，kal，kal，l Lorge kal，l Lorge norked kal nording kal，like norge kal，like norge kal，like norge kal，like norge kal，like nordection，live nordection like nording kal。 [2020]。此外，我们得出了一种计算此类幸存者集的Hausdorff维度以及某些分叉集的结果的方法。此外，通过取代中间体$β$转换的幸存者工会，可以获得一类重要的集合，这些集合在公制数理论中产生，即在非构成基础中的一组差不多的数字集。我们证明，在假设基本的符号空间是有限类型的情况下，这些差不多的数字集中在Schmidt游戏的意义上获胜，因此具有可数的交点属性，从而扩展了Hu和Yu [2014]，Tseng [2009]和Färm，Persson和Schmeleling [2010]。
We show that every intermediate $β$-transformation is topologically conjugate to a greedy $β$-transformation with a hole at zero, and provide a counterexample illustrating that the correspondence is not one-to-one. This characterisation is employed to (1) build a Krieger embedding theorem for intermediate $β$-transformation, complementing the result of Li, Sahlsten, Samuel and Steiner [2019], and (2) obtain new metric and topological results on survivor sets of intermediate $β$-transformations with a hole at zero, extending the work of Kalle, Kong, Langeveld and Li [2020]. Further, we derive a method to calculate the Hausdorff dimension of such survivor sets as well as results on certain bifurcation sets. Moreover, by taking unions of survivor sets of intermediate $β$-transformations one obtains an important class of sets arising in metric number theory, namely sets of badly approximable numbers in non-integer bases. We prove, under the assumption that the underlying symbolic space is of finite type, that these sets of badly approximable numbers are winning in the sense of Schmidt games, and hence have the countable intersection property, extending the results of Hu and Yu [2014], Tseng [2009] and Färm, Persson and Schmeling [2010].
