Spectral properties of a class of Moran measures on R2

Abstract

Given a pair $(R,D)$, where $R={\left\{R_i\right\}}_{i=1}^{\infty }$ is a sequence of expanding matrix (i.e., all the eigenvalues of $R_i$ have modulus strictly greater than 1), and $D={\left\{D_i\right\}}_{i=1}^{\infty }\subseteq Z^2$. It is well known that there exists an infinite convolution generated by $(R,D)$ which satisfies ${\mu }_{R,D}≔{\delta }_{R_1^{-1}{D}_1}\ast {\delta }_{{\left(R_2{R}_1\right)}^{-1}{D}_2}\ast \cdots ,$we say that ${\mu }_{R,D}$ is a Moran measure if it convergent to a probability measure with compact support in a weak sense, where ${\delta }_E=\frac{1}{\#E}{\sum }_{d\in E}{\delta }_d$ is the uniformly discrete measure on $E$. In this paper, we consider the spectral properties of the Moran measure ${\mu }_{R,D}$ with $R=\left(\begin{array}{cc}{b}_1& 0\\ 0& b_2\end{array}\right)$, and $\#D_i=p$, where $1<b_1,b_2\in R$ and $p$ is a prime number, $\underset{i\in N,d\in D_i}{sup}\left|d\right|<\infty$. Let $B\left(p\right):\equiv \left\{\frac{1}{p}\left(\frac{j}{k}\right):1\le j,k\le p-1\right\}$ and $Z\left({\hat{\delta }}_E\right)=\{x\in R^2:{\hat{\delta }}_E\left(x\right)=0\}$. We show that under the conditions that $Z\left({\hat{\delta }}_{D_i}\right)=\bigcup _{k=1}^{\phi \left(i\right)}{Z}_k\left(p\right)$for some $\phi \left(i\right)\in N$, where $Z_k\left(p\right)\subseteq B\left(p\right)+Z^2$ with $(Z_k\left(p\right)-Z_k\left(p\right))\setminus Z^2\subseteq Z_k\left(p\right)$, and $Z_k\left(p\right)⁄\subseteq B\left(q\right)+Z^2$ for $0<q<p$, $1\le k\le \phi \left(i\right)$, $\{x\in {[0,1)}^2:\left|{\hat{\delta }}_{D_i}\left(x\right)\right|=1\}=\left\{\left(\frac{0}{0}\right)\right\}$. Then ${\mu }_{R,D}$ is a spectral measure if and only if $R=\left(\begin{array}{cc}p{k}_1& 0\\ 0& p{k}_2\end{array}\right)$ for some $k_1,k_2\in N$. This extends the results of Sierpinski-type measure considered by Dai et.al [ACHA,2021]. Also some sufficient conditions for ${\mu }_{R,D}$ being a spectral measure when $p$ is not a prime number are given.