In this paper, we study the spectral property of a class of self-affine measures ${\mu }_{R,\mathcal{D}}$ on ${\mathbb{R}}^2$ generated by the iterated function system ${\{{\varphi }_d(\cdot )=R^{-1}(\cdot +d)\}}_{d\in \mathcal{D}}$ associated with the real expanding matrix $R=\left(\begin{array}{cc}\hfill b_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill b_2\hfill \end{array}\right)$ and the digit set $\mathcal{D}=\{\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{c}1\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 1\end{array}\right)\}$. We show that ${\mu }_{R,\mathcal{D}}$ is a spectral measure if and only if $3|b_i$, $i=1,2$. This extends the result of Deng and Lau [J. Funct. Anal., 2015], where they considered the case $b_1=b_2$. And we also give a tree structure for any spectrum of ${\mu }_{R,\mathcal{D}}$ by providing a decomposition property on it.

中文翻译：

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自仿射Sierpinski型测度的谱 ${\mathbb{[R}}^2$

本文研究了一类自仿射测度的光谱特性 ${\mu }_{\mathrm{[R}，\mathcal{d}}$ 上 ${\mathbb{[R}}^2$ 由迭代功能系统生成 ${\{{\varphi }_d（\cdot ）={\mathrm{[R}}^{-\mathrm{1个}}（\cdot +d）\}}_{d\in \mathcal{d}}$ 与实扩展矩阵相关 $\mathrm{[R}=（\begin{array}{cc}\hfill b_{\mathrm{1个}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill b_2\hfill \end{array}）$ 和数字集 $\mathcal{d}=\{（\begin{array}{c}0\\ 0\end{array}），（\begin{array}{c}\mathrm{1个}\\ 0\end{array}），（\begin{array}{c}0\\ \mathrm{1个}\end{array}）\}$。我们证明${\mu }_{\mathrm{[R}，\mathcal{d}}$ 是光谱测量值，当且仅当 $3|b_{\mathrm{一世}}$， $\mathrm{一世}=\mathrm{1个}，2$。这扩展了邓和刘的结果[J．功能 [Anal。，2015]，$b_{\mathrm{1个}}=b_2$。我们还给出了任何频谱的树形结构${\mu }_{\mathrm{[R}，\mathcal{d}}$ 通过提供分解特性。