On spectral eigenmatrix problem for the planar self-affine measures with three digits

Let \(\mu _{M,D}\) ${\mu }_{M,D}$ be a self-affine measure generated by an iterated function systems \(\{\phi _d(x)=M^{-1}(x+d)\ (x\in \mathbb {R}^2)\}_{d\in D}\) ${\{{\varphi }_d(x)=M^{-1}(x+d)(x\in {\mathbb{R}}^2)\}}_{d\in D}$ , where \(M\in M_2(\mathbb {Z})\) $M\in M_2(\mathbb{Z})$ is an expanding integer matrix and \(D = \{(0,0)^t,(1,0)^t,(0,1)^t\}\) $D=\{{(0,0)}^t,{(1,0)}^t,{(0,1)}^t\}$ . In this paper, we study the spectral eigenmatrix problem of \(\mu _{M,D}\) ${\mu }_{M,D}$ , i.e., we characterize the matrix R which \(R\Lambda \) $R\mathrm{\Lambda }$ is also a spectrum of \(\mu _{M,D}\) ${\mu }_{M,D}$ for some spectrum \(\Lambda \) $\mathrm{\Lambda }$ . Some necessary and sufficient conditions for R to be a spectral eigenmatrix are given, which extends some results of An et al. (Indiana Univ Math J, 7(1): 913–952, 2022). Moreover, we also find some irrational spectral eigenmatrices of \(\mu _{M,D}\) ${\mu }_{M,D}$ , which is different from the known results that spectral eigenmatrices are rational.