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

Let $\mu$ be a self-similar measure generated by iterated function system of four maps of equal contraction ratio $0 \lt \rho \lt 0$. We study when μ is a spectral measure which means that it admits an exponential orthonormal basis ${\lbrace e^{2\ \pi i \lambda x}\rbrace}_{\lambda \in \Lambda}$ 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 (Theorem 1.7). Moreover, the case allows us to propose a modified Łaba-Wang conjecture concerning when the self-similar measures are spectral in general cases.

Keywords

Hadamard triples, self-similar measures, spectral measures

2010 Mathematics Subject Classification

28A80, 42B10, 42C30

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Dedicated to the memory of Professor Ka-Sing Lau

Received 18 September 2022

Accepted 5 May 2023

Published 10 July 2024