Spectrum of self-affine measures on the Sierpinski family

Abstract

In this study, a spectrum $\Lambda$ for the integral Sierpinski measures $\mu _{M, D}$ with the digit set $D= \left\{ \begin{pmatrix} 0\\ 0 \end{pmatrix}, \begin{pmatrix} 1\\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix}\right\}$ is derived for a $2 \times 2$ diagonal matrix M with entries as $3\ell _1$ and $3\ell _4$ and for off-diagonal matrix M with both the off-diagonal entries as $3\ell$ where, $\ell ,\ell _1,\ell _4 \in {\mathbb {Z}}{\setminus }{\{0\}}$. Additionally, the spectrum of $\mu _{M, D}$ for a given M and a generalized digit set D is also examined. The spectrum of self-affine measures $\mu _{M, D}$ on spatial Sierpinski gasket is obtained when M is diagonal matrix with entries $\ell _i \in 2{\mathbb {Z}}\setminus {\{0\}}$, sign of $\ell _i$’s are same and $D=\{0, e_1, e_2, e_3\}$, where $e_i's$ are the standard basis in ${\mathbb {R}}^3$. Further, the spectrum of $\mu _{M, D}$ for some off-diagonal $3\times 3$ matrices is also found.