Abstract:

Let $Q=\begin{pmatrix} b & 0\\ 0 & b \end{pmatrix}$ be an integer expanding matrix and let $\mathcal{D}=\left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix},\begin{pmatrix} 1 \\ 0 \end{pmatrix},\begin{pmatrix} 0 \\ 1 \end{pmatrix},\begin{pmatrix} -1 \\ -1 \end{pmatrix} \right\}$ be a four element digit set. We considered the spectral structure of self-similar measure $\mu_{Q,\mathcal{D}}$ which generated by an integer expanding matrix $Q$ and a four element digits $\mathcal{D}$. It is well known that $\mu_{Q,\mathcal{D}}$ is a spectral measure if and only if $b=2q$ for some $q\in\mathbb{N}$. The spectrum for this spectral measure is the following set

where $\mathcal{C}_{q}=q\left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix},\,\,\begin{pmatrix} 1 \\ 0 \end{pmatrix},\,\,\begin{pmatrix} 0 \\ 1 \end{pmatrix},\,\,\begin{pmatrix} 1 \\ 1 \end{pmatrix} \right\}$. In this paper, we investigate the structure of the maximum orthogonal set of $\mu_{Q,\mathcal{D}} $ through the maximum tree mapping and based on this, the relevant issues of its spectral eigenmatrix were discussed.

Key words: maximum tree mapping, spectral eigenmatrix, spectral measure, spectral structure, self-similar measure