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The spectrality of Cantor-Moran measure and Fuglede's Conjecture
- Publication Year :
- 2023
-
Abstract
- Let $\{(p_n, \mathcal{D}_n, L_n)\}$ be a sequence of Hadamard triples on $\mathbb{R}$. Suppose that the associated Cantor-Moran measure $$ \mu_{\{p_n,\mathcal{D}_n\}}=\delta_{p_1^{-1}\mathcal{D}_1}\ast\delta_{(p_2p_1)^{-1}\mathcal{D}_2}\ast\cdots, $$ where $\sup_n\{|p_n^{-1}d|:d\in \mathcal{D}_n\}<\infty$ and $\sup\#\mathcal{D}_n<\infty$. It has been observed that the spectrality of $\mu_{\{p_n,D_n\}}$ is determined by equi-positivity. A significant problem is what kind of Moran measures can satisfy this property. In this paper, we introduce the conception of \textit{Double Points Condition Set} (\textit{DPCS}) to characterize the equi-positivity equivalently. As applications of our characterization, we show that all singularly continuous Cantor-Moran measures are spectral. For the absolutely continuous case, we study Fuglede's Conjecture on Cantor-Moran set. We show that the equi-positivity of $\mu_{\{p_n,D_n\}}$ implies the tiling of its support, and the reverse direction holds under certain conditions.<br />Comment: 34 pages
- Subjects :
- Mathematics - Functional Analysis
42C05, 42A85 (Primary) 28A78, 28A80 (Secondary)
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2305.12135
- Document Type :
- Working Paper