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Spectrality of generalized Sierpinski-type self-affine measures
- Publication Year :
- 2020
-
Abstract
- For an expanding integer matrix $M\in M_2(\mathbb{Z})$ and an integer digit set $D=\{(0,0)^t,(\alpha_1,\alpha_2)^t,(\beta_1,\beta_2)^t\}$ with $\alpha_1\beta_2-\alpha_2\beta_1\neq0$, let $\mu_{M,D}$ be the Sierpinski-type self-affine measure defined by $\mu_{M,D}(\cdot)=\frac{1}{3}\sum_{d\in D}\mu_{M,D}(M(\cdot)-d)$. In [5.36], the authors separately investigated the spectral property of the measure $\mu_{M,D}$ in the case of $\det(M)\notin 3\mathbb{Z}$ or $\alpha_1\beta_2-\alpha_2\beta_1\notin 3\mathbb{Z}$. In this paper, we consider the remaining case where $\det(M)\in 3\mathbb{Z}$ and $\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb{Z}$, and give the necessary and sufficient conditions for $\mu_{M,D}$ to be a spectral measure. This completely settles the spectrality of the Sierpinski-type self-affine measure $\mu_{M,D}$.
- Subjects :
- Mathematics - Classical Analysis and ODEs
Mathematics - Functional Analysis
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2010.14724
- Document Type :
- Working Paper