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

TL;DR

This paper classifies all spectral measures generated by four equal-contraction self-similar systems and discusses implications for the broader conjecture on spectral self-similar measures.

Contribution

It provides a complete classification of spectral measures for four-map self-similar systems and proposes a modified conjecture on spectrality.

Findings

Complete classification of spectral four-map self-similar measures

Identification of new cases where measures are spectral

Proposal of a modified { extL}aba-Wang conjecture

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 {\L}aba-Wang conjecture concerning when the self-similar measures are spectral in general cases.