自仿射测度的傅里叶变换
Fourier transform of self-affine measures
摘要：
假设$F$是$\mathbb{R}^d$，$d\geq 2$上的一个自仿射集，它不是一个单例集，与仿射压缩$f_j = A_j + b_j$，$A_j \in \mathrm{GL}(d,\mathbb{R})$，$b_j \in \mathbb{R}^d$，$j \in \mathcal{A}$相关联，对于某个有限的$\mathcal{A}$ .证明了若由矩阵$A_j$，$j \in \mathcal{A}$生成的群$Γ$构成$\mathrm{GL}(d,\mathbb{R})$的一个邻近且完全不可约的子群，则$F$上的任何自仿射测度$μ= \sum p_j f_j μ$，$\sum p_j = 1$，$0 < p_j < 1$，$j \in \mathcal{A}$是一个Rajchman测度：傅里叶变换$\widehatμ(ξ) \to 0$为$|ξ| \to \infty$ .作为一个应用，这表明，自仿射集的近端和完全不可约的线性部分是集的矩形重的多重三角级数。此外，如果$Γ$的Zeroki闭包是Zeroki拓扑中的实分裂李群，则$\widehatμ(ξ)$在无穷远处有幂衰减。因此，$μ$是$L^p$对所有$1 < p < \infty$的改进，并且$F$具有正傅立叶维数。在$d = 2,3$维上，$Γ$的不可约性和$Γ$在$\mathrm{PGL}(d,\mathbb{R})$中的象的非紧性足以使$\widehatμ$的幂衰减。证明是基于定量更新定理的随机游动的领域$\mathbb{S}^{d-1}$。
英文摘要：
Suppose $F$ is a self-affine set on $\mathbb{R}^d$, $d\geq 2$, which is not a singleton, associated to affine contractions $f_j = A_j + b_j$, $A_j \in \mathrm{GL}(d,\mathbb{R})$, $b_j \in \mathbb{R}^d$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$. We prove that if the group $Γ$ generated by the matrices $A_j$, $j \in \mathcal{A}$, forms a proximal and totally irreducible subgroup of $\mathrm{GL}(d,\mathbb{R})$, then any self-affine measure $μ= \sum p_j f_j μ$, $\sum p_j = 1$, $0 < p_j < 1$, $j \in \mathcal{A}$, on $F$ is a Rajchman measure: the Fourier transform $\widehatμ(ξ) \to 0$ as $|ξ| \to \infty$. As an application this shows that self-affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series. Moreover, if the Zariski closure of $Γ$ is connected real split Lie group in the Zariski topology, then $\widehatμ(ξ)$ has a power decay at infinity. Hence $μ$ is $L^p$ improving for all $1 < p < \infty$ and $F$ has positive Fourier dimension. In dimension $d = 2,3$ the irreducibility of $Γ$ and non-compactness of the image of $Γ$ in $\mathrm{PGL}(d,\mathbb{R})$ is enough for power decay of $\widehatμ$. The proof is based on quantitative renewal theorems for random walks on the sphere $\mathbb{S}^{d-1}$.
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