Spectral property of self-affine measures on

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• 作者：Jing-Cheng Liu^a ; ^{liujingcheng11@126.com" class="auth_mail" title="E-mail the corresponding author} ; Jun Jason Luo^b ; ^{jun.luo@cqu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 42C05 ; secondary ; 28A80
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：272
• 期：2
• 页码：599-612
• 全文大小：347 K

文摘

We study spectral properties of the self-affine measure class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si116.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=c3b578b77d2661841fe0bebf5a29745f" title="Click to view the MathML source">μ_M,Dclass="mathContainer hidden">class="mathCode">${\mu }_{M,\mathcal{D}}$ generated by an expanding integer matrix class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si264.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=896df5673b3e68e825c1b43849429d32" title="Click to view the MathML source">M∈M_n(Z)class="mathContainer hidden">class="mathCode">$M\in M_n(\mathbb{Z})$ and a consecutive collinear digit set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si131.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=dfa9cb06d4a31d6498b89f37c47fb1d3" title="Click to view the MathML source">D={0,1,…,q−1}vclass="mathContainer hidden">class="mathCode">$\mathcal{D}=\{0,1,\dots ,q-1\}v$ where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si115.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=bfef620d8742554af638bd68a685a048" title="Click to view the MathML source">v∈Zⁿ∖{0}class="mathContainer hidden">class="mathCode">$v\in {\mathbb{Z}}^n\setminus \{0\}$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si52.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=6cef4677b5a4a6cc407922e007352eff" title="Click to view the MathML source">q≥2class="mathContainer hidden">class="mathCode">$q\ge 2$ is an integer. Some sufficient conditions for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si116.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=c3b578b77d2661841fe0bebf5a29745f" title="Click to view the MathML source">μ_M,Dclass="mathContainer hidden">class="mathCode">${\mu }_{M,\mathcal{D}}$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si116.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=c3b578b77d2661841fe0bebf5a29745f" title="Click to view the MathML source">μ_M,Dclass="mathContainer hidden">class="mathCode">${\mu }_{M,\mathcal{D}}$. Our study generalizes the one dimensional results proved by Dai, et al.  and .