一类自仿测度的非谱性
首发时间:2019-11-07
摘要:假设三元数字集~$D=\left\{\left(\begin{array}{c}0\\0\end{array}\right),\left(\begin{array}{c}\alpha\\\beta\end{array}\right),\left(\begin{array}{c}\gamma\\\eta\end{array}\right)\right\}\subset \mathbb{Z}^{2}$和二阶整数扩张矩阵~$M\in M_{2}(\mathbb{Z})$~满足~$\alpha\eta-\beta\gamma \notin 3\mathbb{Z}$~和~$\det(M)\in3\mathbb{Z}$.设~$\mu_{M,D}$~是~$M$~和~$D$~生成的自仿测度. 在这篇文章中, 我们给出~$L^{2}(\mu_{M,D})$~中最多包含有限个正交指数函数的充分条件, 特别地, 在这个条件下,~$\mu_{M,D}$~是非谱测度.
关键词: 自仿测度; 非谱测度; 正交指数函数.
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Non-spectrality of Some Self-affine Measures
Abstract:Let $D=\left\{\left(\begin{array}{c}0\\0\end{array}\right),\left(\begin{array}{c}\alpha\\\beta\end{array}\right),\left(\begin{array}{c}\gamma\\\eta\end{array}\right)\right\}\subset \mathbb{Z}^{2}$ be a three-element digit set and $M\in M_{2}(\mathbb{Z})$ be a $2\times 2$ expanding integer matrix satisfying $\alpha\eta-\beta\gamma \notin 3\mathbb{Z}$ and $\det(M)\in 3\mathbb{Z}$. Let $\mu_{M,D}$ be the self-affine measure generated by $M$ and $D$. In this paper, we give a sufficient condition such that $L^{2}(\mu_{M,D})$ contains at most finite number of orthogonal exponential functions, in particular, under this condition, $\mu_{M,D}$ is non-spectral measure.
Keywords: Self-affine measure Non-spectral measure Orthogonal exponential functions.
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