AI 中文总结
研究与乘积形式哈达玛三元组相关的自相似测度的谱特征值问题，引入乘子集T*，证明特定谱特征子空间基数为连续统，得出四位数自相似测度谱特征值及谱特征值集的相关结论。
AI 中文摘要
此前，An表明由乘积形式哈达玛三元组生成的自相似测度μ是谱测度。本文研究其谱特征值问题。若存在μ的谱Λ使得对每个a∈A，aΛ都是μ的谱，则集合A⊂R称为μ的谱特征值集。引入乘积形式哈达玛乘子集T*，证明对任意s∈[0,log#D/log N]，谱特征子空间V(s)(μN,D,T*)的基数为连续统。此结果表明对于四位数自相似测度，实数t是谱特征值当且仅当t∈{u/v:u,v∈2Z+1}，且任意子集S是谱特征值集当且仅当S⊂t−1(2Z+1)对某个t∈2Z+1成立。
英文摘要
Previously, An \cite{AL01} showed that the self-similar measure $μ$ generated by a product-form Hadamard triple is a spectral measure. In this paper, we study its spectral eigenvalue problem. A set $A\subset\mathbb R$ is called a spectral eigenvalue set of $μ$ if there exists a spectrum $Λ$ of $μ$ such that $aΛ$ is a spectrum of $μ$ for every $a\in A$. We introduce the Product-form Hadamard multiplier set $\mathcal{T}_*$, and prove that for any $s\in [0,\frac{\log \#\mathcal{D}}{\log N}]$, the spectral eigensubspace $$V^{(s)}(μ_{N,\mathcal{D}},\mathcal{T}_*):=\{Λ:t Λ\text{ is a spectrum of }μ\text{ for all }t \in\mathcal{T}_* \text{ and } \dim_{Be}(Λ)=s\}$$ has the cardinality of the continuum. This result allows us to show that for the four-digit self-similar measures, a real number $t$ is a spectral eigenvalue if and only if $t \in \left\{\frac{u}{v}:u,v\in 2\mathbb{Z}+1\right\}$. And for any subset $S$ of $\mathbb{R}$ is a spectral eigenvalue set if and only if $S \subset t^{-1} (2\mathbb{Z}+1)$ for some $t\in 2\mathbb{Z}+1$.
