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.