On the spectra of a class of Moran measures

Abstract

Abstract Let { A n } n = 1 ∞ {\{A_{n}\}_{n=1}^{\infty}} be a sequence of expanding matrices with A n ∈ M 2 ⁢ ( ℤ ) {A_{n}\in M_{2}(\mathbb{Z})} , and let { D n } n = 1 ∞ {\{D_{n}\}_{n=1}^{\infty}} be a sequence of three-element digit sets with { x ∈ ( 0 , 1 ) 2 : ∑ d ∈ D n e 2 ⁢ π ⁢ i ⁢ 〈 d , x 〉 = 0 } = { ± 1 3 ⁢ ( 1 , i ) t } {\{x\in(0,1)^{2}:\sum_{d\in D_{n}}{e^{2\pi i\langle d,x\rangle}}=0\}=\{\pm% \frac{1}{3}(1,i)^{t}\}} , i ∈ { 1 , 2 } {i\in\{1,2\}} . The associated Moran measure generated by the infinite convolution μ { A n } , { D n } = δ A 1 - 1 ⁢ D 1 * δ A 1 - 1 ⁢ A 2 - 1 ⁢ D 2 * δ A 1 - 1 ⁢ A 2 - 1 ⁢ A 3 - 1 ⁢ D 3 * ⋯ . \mu_{\{A_{n}\},\{D_{n}\}}=\delta_{A_{1}^{-1}D_{1}}*\delta_{A_{1}^{-1}A_{2}^{-1% }D_{2}}*\delta_{A_{1}^{-1}A_{2}^{-1}A_{3}^{-1}D_{3}}*\cdots. In this paper, we give some necessary and sufficient conditions for μ { A n } , { D n } {\mu_{\{A_{n}\},\{D_{n}\}}} to be a spectral measure under some suitable conditions on A n {A_{n}} and D n {D_{n}} .