The continuity of Beurling density and Beurling dimension of spectra of a class of self-affine spectral measures
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918165456355328 |
|---|---|
| author | Chen, Zi-Yun Wu, Zhi-Yi Zhang, Min-Min |
| author_facet | Chen, Zi-Yun Wu, Zhi-Yi Zhang, Min-Min |
| contents | It is well-known that the Beurling dimension of the spectra of certain singularly continuous spectral measures possesses an intermediate property. In this paper, we establish that for a class of self-affine spectral measures $μ$, both the Beurling dimension and Beurling density of their spectra attain full flexibility simultaneously. Specifically, for any $t\in (0,\dim_H^w(\supp(μ))]$ and $s\in [0,\infty)$, there exists a spectrum $Λ:=Λ_{t,s}$ of $μ$ satisfying \[\dim_{Be}(Λ)=t\quad\text{and}\quad D_{t}^+(Λ)=s\] where $\dim_H^w$ denotes the pseudo Hausdorff dimension, $\dim_{Be}$ denotes the Beurling dimension and $D_{t}^+$ denotes the $t$-Beurling density. These results provide new insights into the structure of the spectra for a singularly continuous spectral measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19187 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The continuity of Beurling density and Beurling dimension of spectra of a class of self-affine spectral measures Chen, Zi-Yun Wu, Zhi-Yi Zhang, Min-Min Classical Analysis and ODEs Dynamical Systems It is well-known that the Beurling dimension of the spectra of certain singularly continuous spectral measures possesses an intermediate property. In this paper, we establish that for a class of self-affine spectral measures $μ$, both the Beurling dimension and Beurling density of their spectra attain full flexibility simultaneously. Specifically, for any $t\in (0,\dim_H^w(\supp(μ))]$ and $s\in [0,\infty)$, there exists a spectrum $Λ:=Λ_{t,s}$ of $μ$ satisfying \[\dim_{Be}(Λ)=t\quad\text{and}\quad D_{t}^+(Λ)=s\] where $\dim_H^w$ denotes the pseudo Hausdorff dimension, $\dim_{Be}$ denotes the Beurling dimension and $D_{t}^+$ denotes the $t$-Beurling density. These results provide new insights into the structure of the spectra for a singularly continuous spectral measure. |
| title | The continuity of Beurling density and Beurling dimension of spectra of a class of self-affine spectral measures |
| topic | Classical Analysis and ODEs Dynamical Systems |
| url | https://arxiv.org/abs/2510.19187 |