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arXiv · 2209.05619

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

Li-Xiang An·Xinggang He·Chun-Kit Lai
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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.

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Li-Xiang An, Xinggang He, Chun-Kit Lai. 2022-09-12. Classification of spectral self-similar measures with four-digit elements. https://arxiv.org/abs/2209.05619

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