Spectra of the Sierpiński type spectral measure and their Beurling dimensions

Abstract

The Sierpiński type measures are an important class of self-affine measures studied by specialists in geometric measure theory, dynamical systems and probability. In this paper, we investigate the structure of the spectra for the Sierpiński type spectral measure ${\mu }_{A,D}$. We first give a sufficient and necessary condition for the family of exponential functions $\{e^{-2\pi i〈\lambda ,x〉}:\lambda \in \Lambda \}$ to be a maximal orthogonal set in $L^2\left({\mu }_{A,D}\right)$. Based on this result, we construct a class of regular spectra of ${\mu }_{A,D}$. Furthermore, we analyze the Beurling dimensions of the spectra and obtain the optimal upper bound of Beurling dimensions of all spectra, which is in stark contrast with the case of self-similar spectral measures. As an application of our results, we obtain an intermediate property about the Beurling dimension of the spectra.