Spectrality and non-spectrality of self-affine measures with five-element digit sets on $\R^{2}$

Abstract

Let $M=\begin{pmatrix} \rho & 0\\ 0 & \rho \end{pmatrix}$ be an expanding real matrix, and let $\mathcal{D}=\left\{\begin{pmatrix} 0\\ 0 \end{pmatrix},\,\,\,\begin{pmatrix} 1\\ 0 \end{pmatrix}, \,\,\,
\begin{pmatrix} 0\\ 1 \end{pmatrix}, \,\,\,
\begin{pmatrix} 1\\ -1 \end{pmatrix}, \,\,\,
\begin{pmatrix} -1\\ 1 \end{pmatrix} \right\}$ be a digit set. In this paper, we mainly study the properties of spectra of self-affine measure $\mu_{M,\mathcal{D}}$ generated by $M$ and $\mathcal{D}$. We showed that $\mu_{M,\mathcal{D}}$ is a spectral measure if and only if $5\mid \rho$. Furthermore, by extending the maximal mapping to plane, we gives a characterization for $E(\Lambda)$ to be a maximal orthogonal family in $L^{2}(\mu_{M,\mathcal{D}})$. Based on these, we also obtained some sufficient conditions for the maximal orthogonal set to be an orthogonal basis of $L^{2}(\mu_{M,\mathcal{D}})$.