A Class of Spectral Moran Measures Generated by the Compatible Tower

Let \(\big \{(M_n^{-1}B_n,C_n)\big \}_{n=1}^{\infty }\) $\{(M_n^{-1}{B}_n,C_n){\}}_{n=1}^{\infty }$ be a compatible tower on \({\mathbb R}^d\) ${\mathbb{R}}^d$ and let \(\mu _{\{M_n\},\{B_n\}}\) ${\mu }_{\{M_n\},\{B_n\}}$ be the Moran measure generated by infinite convolutions of discrete measures induced by them. In this paper, we first prove that under certain situations, the compatible tower condition can ensure that \(\mu _{\{M_n\},\{B_n\}}\) ${\mu }_{\{M_n\},\{B_n\}}$ is a spectral measure, that is the Hilbert space \(L^2(\mu _{\{M_n\},\{B_n\}})\) $L^2({\mu }_{\{M_n\},\{B_n\}})$ admits an exponential orthonormal basis. Furthermore, if we restrict \(\big \{M_n,B_n\big \}_{n=1}^{\infty }\) $\{M_n,B_n{\}}_{n=1}^{\infty }$ to be a class of generalized Sierpinski-type family, then we obtain that the existence of compatible tower and the spectrality of \(\mu _{\{M_n\},\{B_n\}}\) ${\mu }_{\{M_n\},\{B_n\}}$ are equivalent.