论文标题
广义sierpinski型自我措施的光谱
Spectrality of generalized Sierpinski-type self-affine measures
论文作者
论文摘要
对于M_2（\ Mathbb {Z}）$中的扩展整数矩阵$ m \和一个整数数字集$ d = \ {（0,0,0）^t，（α_1，α_2）^t，（β_1，β_1，β_1，β_2） $μ_{m，d} $是由$μ_{m，d}定义的sierpinski-type自动措施（\ cdot）= \ frac {1} {1} {3} \ sum_ {d \ in d}μ_{m，d}（m，d}（m，d}（m（\ cdot））$。在[5.36]中，作者在$ \ det（m）\ notin 3 \ mathbb {z} $或$α_1β_1β_2-α_2-α_2β_2β_1\ notin 3 \ notin 3 \ mathbb {z} $中分别研究了该度量$μ_{m，d} $的光谱特性。在本文中，我们考虑了剩余的情况，其中$ \ det（m）\在3 \ mathbb {z} $和$α_1β_2-α_2β_2β_1\ in 3 \ mathbb {z} $中，并提供$μ__{m，d} $的必要条件。这完全解决了sierpinski型自动措施$μ_{m，d} $的光谱。
For an expanding integer matrix $M\in M_2(\mathbb{Z})$ and an integer digit set $D=\{(0,0)^t,(α_1,α_2)^t,(β_1,β_2)^t\}$ with $α_1β_2-α_2β_1\neq0$, let $μ_{M,D}$ be the Sierpinski-type self-affine measure defined by $μ_{M,D}(\cdot)=\frac{1}{3}\sum_{d\in D}μ_{M,D}(M(\cdot)-d)$. In [5.36], the authors separately investigated the spectral property of the measure $μ_{M,D}$ in the case of $\det(M)\notin 3\mathbb{Z}$ or $α_1β_2-α_2β_1\notin 3\mathbb{Z}$. In this paper, we consider the remaining case where $\det(M)\in 3\mathbb{Z}$ and $α_1β_2-α_2β_1\in 3\mathbb{Z}$, and give the necessary and sufficient conditions for $μ_{M,D}$ to be a spectral measure. This completely settles the spectrality of the Sierpinski-type self-affine measure $μ_{M,D}$.
