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

Abstract

Let \(\mu _{M,D}\) be a self-affine measure generated by an iterated function systems \(\{\phi _d(x)=M^{-1}(x+d)\ (x\in \mathbb {R}^2)\}_{d\in D}\), where \(M\in M_2(\mathbb {Z})\) is an expanding integer matrix and \(D = \{(0,0)^t,(1,0)^t,(0,1)^t\}\). In this paper, we study the spectral eigenmatrix problem of \(\mu _{M,D}\), i.e., we characterize the matrix R which \(R\Lambda \) is also a spectrum of \(\mu _{M,D}\) for some spectrum \(\Lambda \). 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 \(\mu _{M,D}\), which is different from the known results that spectral eigenmatrices are rational.