Spectrality of certain Moran measures in two-dimension

• 1、School of Mathematics, Hunan University, Changsha 410082

Abstract：Let $R_k=\begin{pmatrix}a_k &0\\0&b_k\end{pmatrix}$ be an expanding positive integer matrix, and let $D_{k}=$ $\{0,1,\cdots,q_{k}-1\}(\tau, \tau')^t$ be a digit set, where $\tau,\; \tau',\; q_k $ are positive integers, and $q_k>1$.In this paper, we study the spectrality of the Moran measure $\mu_{\{R_k\}\{D_k\}}$ generated by$$\mu_{\{R_k\}\{D_k\}}:=\delta_{R_1^{-1}D_1}\ast\delta_{(R_2R_1)^{-1}D_2}\ast\cdots\ast\delta_{(R_k\cdots R_2R_1)^{-1}D_k}\ast\cdots.$$We proved that if~$q_{k}\tau \mid a_{k}\;( k \in {\mathbb{Z}}^{+})$~or ~$q_{k}\tau' \mid b_{k}\;(k \in {\mathbb{Z}}^{+})$, then $\mu_{\{R_k\}\{D_k\}}$ is spectral measure, and that if ~$\gcd(\tau \tau',q_{k})=1,q_{k} \mid \frac{a_{k}b_{k}\tau \tau'}{\gcd(a_{k}\tau',b_{k}\tau)},\;l=\lim \sup_{k\rightarrow\infty}q_{k}(\frac{\tau}{a_{k}}+\frac{\tau'}{b_{k}})<1$, then $\mu_{\{R_k\}\{D_k\}}$ is spectral measure.

Keywords： Moran measure Spectral measure Spectra