论文标题
自符号措施的尖端正态性和傅立叶衰减
Pointwise normality and Fourier decay for self-conformal measures
论文作者
论文摘要
令$φ$为$ c^{1+γ} $ smooth ifs in $ \ mathbb {r} $，其中$γ> 0 $。我们在衍生物合过程中提供温和的条件，以确保在绝对正常的$ x $上支持每个自相矛盾的措施。也就是说，对于整数$ p \ geq 2 $序列$ \ lbrace p^k x \ rbrace_ {k \ in \ mathbb {n}} $ equidistributes modulo $ 1 $。因此，我们扩展了Hochman和Shmerkin关于分形中正常数的流行率的几个状态。当$φ$是自相似的时候，我们证明了绝对正常数字的集合在其吸引子中具有完整的Hausdorff尺寸，除非$φ$具有与某些整数$ n \ geq 2 $相关的显式结构。衍生物旋转的这些条件还表明，每个自我形式的措施都是rajchman的措施，即，其傅立叶变换衰减至Infinity $ 0 $。当$φ$是自我相似的并且满足一定的二氧化碳条件时，我们会建立对数衰减速率。
Let $Φ$ be a $C^{1+γ}$ smooth IFS on $\mathbb{R}$, 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 integer $p\geq 2$ the sequence $\lbrace p^k x \rbrace_{k\in \mathbb{N}}$ equidistributes modulo $1$. We thus extend several state of the art results of Hochman and Shmerkin 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 $n\geq 2$. 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.
