${\mathbb R}^n$上自仿射测度的谱性质
Spectral property of self-affine measures on ${\mathbb R}^n$
摘要：
研究了由整数矩阵$M\in M_n(\mathbb{Z})$和连续共线数字集$\mathcal {D}=\{0,1,\dots,q-1\}v$生成的自仿射测度$μ_{M,\mathcal {D}}$的谱性质，其中$v\in \mathbb{Z}^n\setminus\{0\}$和$q\ge 2$均为整数.给出了$μ_{M,\mathcal {D}}$是谱测度或有无穷多个正交指数的充分条件。此外，对于某些特殊情况，我们还得到了$μ_{M,\mathcal {D}}$谱性的一个充要条件。我们的研究推广了Dai，{ \it et al.}（\cite{Dai-He-Lai_2013, Dai-He-Lau_2014}）。
英文摘要：
We study spectral properties of the self-affine measure $μ_{M,\mathcal {D}}$ generated by an expanding integer matrix $M\in M_n(\mathbb{Z})$ and a consecutive collinear digit set $\mathcal {D}=\{0,1,\dots,q-1\}v$ where $v\in \mathbb{Z}^n\setminus\{0\}$ and $q\ge 2$ is an integer. Some sufficient conditions for $μ_{M,\mathcal {D}}$ to be a spectral measure or to have infinitely many orthogonal exponentials are given. Moreover, for some special cases, we can obtain a necessary and sufficient condition on the spectrality of $μ_{M,\mathcal {D}}$. Our study generalizes the one dimensional results proved by Dai, {\it et al.} (\cite{Dai-He-Lai_2013, Dai-He-Lau_2014}).
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