Spectrality of a class of moran measures on $\mathbb{R}^2$
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915451966062592 |
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| author | Liu, Jing-Cheng Liu, Qiao-Qin Luo, Jun Jason Wang, Jia-jie |
| author_facet | Liu, Jing-Cheng Liu, Qiao-Qin Luo, Jun Jason Wang, Jia-jie |
| contents | We investigate spectral properties of planar Moran measures $μ_{\{M_n\},\{D_n\}}$ generated by sequences of expanding matrices $\{M_n\}\subset GL(2,\mathbb{Z})$ and digit sets $\{D_n\}\subset\mathbb{Z}^2$, where each digit set has the form
$$
D_n = \left\{
\begin{pmatrix} 0 \\ 0 \end{pmatrix},
\begin{pmatrix} α_{n_1} \\ α_{n_2} \end{pmatrix},
\begin{pmatrix} β_{n_1} \\ β_{n_2} \end{pmatrix},
\begin{pmatrix} -α_{n_1}-β_{n_1} \\ -α_{n_2}-β_{n_2} \end{pmatrix}
\right\}
$$
satisfying $α_{n_1}β_{n_2}-α_{n_2}β_{n_1} \ne 0 \pmod{2}$. Under the hypotheses $|\det(M_n)| > 4$ for all $n\geq 1$, $\sup_{n\geq 1}\|M_n^{-1}\| < 1$, and $\{D_n\}$ is finite, we establish the following characterization:
$$
μ_{\{M_n\},\{D_n\}} \text{ is a spectral measure} \Longleftrightarrow M_n \in GL(2,2\mathbb{Z}) \text{ for all } n\geq 2.
$$
Furthermore, for the critical case $|\det(M_n)| = 4$, we derive a complete spectral criterion for a significant class of Moran measures through combinatorial analysis of digit sets. These results extend current understanding of spectral self-affine measures to Moran-type constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11200 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectrality of a class of moran measures on $\mathbb{R}^2$ Liu, Jing-Cheng Liu, Qiao-Qin Luo, Jun Jason Wang, Jia-jie Functional Analysis Classical Analysis and ODEs Primary 28A80, Secondary 42B10, 42C05 We investigate spectral properties of planar Moran measures $μ_{\{M_n\},\{D_n\}}$ generated by sequences of expanding matrices $\{M_n\}\subset GL(2,\mathbb{Z})$ and digit sets $\{D_n\}\subset\mathbb{Z}^2$, where each digit set has the form $$ D_n = \left\{ \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} α_{n_1} \\ α_{n_2} \end{pmatrix}, \begin{pmatrix} β_{n_1} \\ β_{n_2} \end{pmatrix}, \begin{pmatrix} -α_{n_1}-β_{n_1} \\ -α_{n_2}-β_{n_2} \end{pmatrix} \right\} $$ satisfying $α_{n_1}β_{n_2}-α_{n_2}β_{n_1} \ne 0 \pmod{2}$. Under the hypotheses $|\det(M_n)| > 4$ for all $n\geq 1$, $\sup_{n\geq 1}\|M_n^{-1}\| < 1$, and $\{D_n\}$ is finite, we establish the following characterization: $$ μ_{\{M_n\},\{D_n\}} \text{ is a spectral measure} \Longleftrightarrow M_n \in GL(2,2\mathbb{Z}) \text{ for all } n\geq 2. $$ Furthermore, for the critical case $|\det(M_n)| = 4$, we derive a complete spectral criterion for a significant class of Moran measures through combinatorial analysis of digit sets. These results extend current understanding of spectral self-affine measures to Moran-type constructions. |
| title | Spectrality of a class of moran measures on $\mathbb{R}^2$ |
| topic | Functional Analysis Classical Analysis and ODEs Primary 28A80, Secondary 42B10, 42C05 |
| url | https://arxiv.org/abs/2412.11200 |