Pointwise normality and Fourier decay for self-conformal measures
Pointwise normality and Fourier decay for self-conformal measures
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DOI：
10.1016/j.aim.2021.108096
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发表时间：
2020-12
影响因子：
1.7
通讯作者：
A. Algom;F. R. Hertz;Zhiren Wang
中科院分区：
文献类型：
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作者：
A. Algom;F. R. Hertz;Zhiren Wang
设Φ是R上的C1 + γ光滑IFS，其中γ> 0.我们提供了温和的条件，确保每一个自我共形措施的导数上循环是绝对正常的点x上的支持。也就是说，对于任何整数p≥ 2，序列{pkx} k∈ N均等分布模1。因此，我们扩展了Hochman和Shmerkin [29]关于分形中正态数流行的几个最新结果。当Φ是自相似的，我们证明了绝对正规数的集合在它的吸引子中具有全Hausdorff维数，除非Φ有一个与某个整数n≥ 2相关联的显式结构。这些关于导数上圈的条件也表明，每个自共形测度是一个Rajchman测度，即它的傅里叶变换在无穷远处衰减到0。当Φ是自相似的，并满足一定的丢番图条件，我们建立一个对数衰减率。
Let Φ be a C 1+ γ smooth IFS on R, where γ> 0. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points x that are absolutely normal. That is, for every integer p≥ 2 the sequence {p k x} k∈ N equidistributes modulo 1. We thus extend several state of the art results of Hochman and Shmerkin [29] about the prevalence of normal numbers in fractals. When Φ is self-similar we show that the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless Φ has an explicit structure that is associated with some integer n≥ 2. These conditions on the derivative cocycle are also shown to imply that every self conformal measure is a Rajchman measure, that is, its Fourier transform decays to 0 at infinity. When Φ is self similar and satisfies a certain Diophantine condition, we establish a logarithmic rate of decay.