Spectrality of Cantor–Moran measures with three-element digit sets

Abstract

Abstract Let 0 < ρ < 1 {0<\rho<1} and let { a j , b j , n j } j = 1 ∞ {\{a_{j},b_{j},n_{j}\}_{j=1}^{\infty}} be a sequence of positive integers with an upper bound. Associated with them, there exists a unique Borel probability measure μ ρ , { 0 , a j , b j } , { n j } {\mu_{\rho,\{0,a_{j},b_{j}\},\{n_{j}\}}} generated by the following infinite convolution of discrete measures: μ ρ , { 0 , a j , b j } , { n j } = δ ρ n 1 ⁢ { 0 , a 1 , b 1 } ∗ δ ρ n 1 + n 2 ⁢ { 0 , a 2 , b 2 } ∗ δ ρ n 1 + n 2 + n 3 ⁢ { 0 , a 3 , b 3 } ∗ ⋯ , \mu_{\rho,\{0,a_{j},b_{j}\},\{n_{j}\}}=\delta_{\rho^{n_{1}}\{0,a_{1},b_{1}\}}% \ast\delta_{\rho^{n_{1}+n_{2}}\{0,a_{2},b_{2}\}}\ast\delta_{\rho^{n_{1}+n_{2}+% n_{3}}\{0,a_{3},b_{3}\}}\ast\cdots, where gcd ⁡ ( a j , b j ) = 1 {\gcd(a_{j},b_{j})=1} for all j ∈ ℕ {j\in{\mathbb{N}}} . In this paper, we show that L 2 ⁢ ( μ ρ , { 0 , a j , b j } , { n j } ) {L^{2}(\mu_{\rho,\{0,a_{j},b_{j}\},\{n_{j}\}})} admits an exponential orthonormal basis if and only if the following two conditions are satisfied: (i) { a j , b j } ≡ { ± 1 } ⁢ ( mod ⁢ 3 ) {\{a_{j},b_{j}\}\equiv\{\pm 1\}~{}(\mathrm{mod}~{}3)} for all j ≥ 1 {j\geq 1} ; (ii) there exists a natural number r such that ρ - r ∈ 3 ⁢ ℕ {\rho^{-r}\in 3{\mathbb{N}}} and n j ∈ r ⁢ ℕ {n_{j}\in r{\mathbb{N}}} for all j ≥ 2 {j\geq 2} .