Abstract Let the 2 × 2 {2\times 2} expanding matrix R k {R_{k}} be an integer Jordan matrix, i.e., R k = diag ⁡ ( r k , s k ) {R_{k}=\operatorname{diag}(r_{k},s_{k})} or R k = J ⁢ ( p k ) {R_{k}=J(p_{k})} , and let D k = { 0 , 1 , … , q k - 1 } ⁢ v {D_{k}=\{0,1,\ldots,q_{k}-1\}v} with v = ( 1 , 1 ) T {v=(1,1)^{T}} and 2 ≤ q k ≤ p k , r k , s k {2\leq q_{k}\leq p_{k},r_{k},s_{k}} for each natural number k. We show that the sequence of Hadamard triples { ( R k , D k , C k ) } {\{(R_{k},D_{k},C_{k})\}} admits a spectrum of the associated Moran measure provided that lim inf k → ∞ ⁡ 2 ⁢ q k ⁢ ∥ R k - 1 ∥ < 1 {\liminf_{k\to\infty}2q_{k}\lVert R_{k}^{-1}\rVert<1} .

中文翻译：

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具有连续和共线数字集的某些 Moran 测度的光谱特性

摘要 令 2 × 2 {2\times 2} 展开矩阵 R k {R_{k}} 为整数 Jordan 矩阵，即 R k = diag ⁡ ( rk , sk ) {R_{k}=\operatorname{diag }(r_{k},s_{k})} 或 R k = J ⁢ ( pk ) {R_{k}=J(p_{k})} ，让 D k = { 0 , 1 , … , qk - 1 } ⁢ v {D_{k}=\{0,1,\ldots,q_{k}-1\}v} 与 v = ( 1 , 1 ) T {v=(1,1)^{T }} 和 2 ≤ qk ≤ pk , rk , sk {2\leq q_{k}\leq p_{k},r_{k},s_{k}} 对于每个自然数 k。我们证明了 Hadamard 三元组的序列 { ( R k , D k , C k ) } {\{(R_{k},D_{k},C_{k})\}} 承认相关的 Moran 测度的频谱假设 lim inf k → ∞ ⁡ 2 ⁢ qk ⁢ ∥ R k - 1 ∥ < 1 {\liminf_{k\to\infty}2q_{k}\lVert R_{k}^{-1}\rVert<1} .