Spectral eigenvalue set of self-similar measures associated with product-form Hadamard triples

Abstract

Previously, An \cite{AL01} showed that the self-similar measure $\mu$ 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 $\mu$ if there exists a spectrum $\Lambda$ of $\mu$ such that $a\Lambda$ is a spectrum of $\mu$ for every $a\in A$. We introduce the Product-form Hadamard multiplier set ${\mathcal{T}}_{\ast }$, and prove that for any $s\in [0,\frac{\log \#\mathcal{D}}{\log N}]$, the spectral eigensubspace $$V^{(s)}({\mu }_{N,\mathcal{D}},{\mathcal{T}}_{\ast }):=\{\Lambda :t\Lambda \text{ is a spectrum of }\mu \text{ for all }t\in {\mathcal{T}}_{\ast }\text{ and }{\dim }_{Be}(\Lambda )=s\}$$ 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 $t\in 2\mathbb{Z}+1$.