Classification of spectral self-similar measures with four-digit elements

Abstract

Let $\mu$ be a self-similar measure generated by iterated function system of four maps of equal contraction ratio $0<\rho <1$. We study when $\mu$ is a spectral measure which means that it admits an exponential orthonormal basis $\{e^{2\pi i\lambda x}{\}}_{\lambda \in \Lambda }$ in $L^2(\mu )$. By combining previous results of many authors and a careful study of some new cases, we completely classify all spectral self-similar measures with four maps. Moreover, the case allows us to propose a modified Łaba-Wang conjecture concerning when the self-similar measures are spectral in general cases.