Spectrality of a class of moran measures on $\mathbb{R}^2$
| Autor: | Liu, Jing-Cheng, Liu, Qiao-Qin, Luo, Jun Jason, Wang, Jia-jie |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $\mu_{\{M_n\},\{D_n\}}$ be a Moran measure on $\mathbb{R}^2$ generated by a sequence of expanding matrices $\{M_n\}\subset GL(2, \mathbb{Z})$ and a sequence of integer digit sets $\{D_n\}$ where $D_n=\left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix},\begin{pmatrix} \alpha_{n_1} \\ \alpha_{n_2} \end{pmatrix},\begin{pmatrix} \beta_{n_1} \\ \beta_{n_2} \end{pmatrix},\begin{pmatrix} -\alpha_{n_1}-\beta_{n_1} \\ -\alpha_{n_2}-\beta_{n_2} \end{pmatrix} \right\}$ with $\alpha_{n_1}\beta_{n_2}-\alpha_{n_2}\beta_{n_1}\notin2\mathbb{Z}$. If $|\det(M_n)|>4$ for $n\geq1$, $\sup\limits_{n\ge 1}\Vert M_n^{-1}\Vert<1$ and $\#\{D_n: n\ge 1\}<\infty$, then we show that $\mu_{\{M_n\},\{D_n\}}$ is a spectral measure if and only if $M_n\in GL(2, 2\mathbb{Z})$ for $n\geq2$. If $|\det(M_n)| =4$ for $n\geq1$, we also establish a necessary and sufficient condition for a class of special Moran measures to be spectral measures. Comment: 19 pages |
| Databáze: | arXiv |
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