Spectrum of self-affine measures on the Sierpinski family

Monatshefte für Mathematik Pub Date : 2024-02-29 DOI:10.1007/s00605-023-01939-7

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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.

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西尔平斯基族上的自阿芬度量谱

Abstract In this study, a spectrum \(\Lambda \) for the integral Sierpinski measures \(\mu _{M, D}\) with the digit set \( D= \left\{ \begin{pmatrix})。0 （end{pmatrix}）, （begin{pmatrix}）1\ 0 （end{pmatrix}）, （begin{pmatrix}）0 1 （end{pmatrix}）。\对于对角矩阵M的对角条目为（3\ell _1）和（3\ell _4）以及非对角矩阵M的非对角条目均为（3\ell \），可以得出（\ell ,\ell_1,\ell_4在{\mathbb{Z}}{setminus}\{0\}}）。此外，我们还研究了给定 M 和广义数字集 D 的 \(\mu _{M, D}\) 的谱。当 M 是对角矩阵时，可以得到空间西尔平斯基垫圈上的自（\ell _i \in 2{\mathbb {Z}}\setminus {\{0\}}\) 的谱、D={0, e_1, e_2, e_3\} （其中 e_i's （）是在 {\mathbb {R}}^3\) 中的标准基础）。此外，还找到了一些非对角\(3\times 3\) 矩阵的\(\mu _{M, D}\) 谱。

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Monatshefte für Mathematik

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