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A Class of Spectral Moran Measures Generated by the Compatible Tower

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Abstract

Let \(\big \{(M_n^{-1}B_n,C_n)\big \}_{n=1}^{\infty }\) be a compatible tower on \({\mathbb R}^d\) and let \(\mu _{\{M_n\},\{B_n\}}\) be the Moran measure generated by infinite convolutions of discrete measures induced by them. In this paper, we first prove that under certain situations, the compatible tower condition can ensure that \(\mu _{\{M_n\},\{B_n\}}\) is a spectral measure, that is the Hilbert space \(L^2(\mu _{\{M_n\},\{B_n\}})\) admits an exponential orthonormal basis. Furthermore, if we restrict \(\big \{M_n,B_n\big \}_{n=1}^{\infty }\) to be a class of generalized Sierpinski-type family, then we obtain that the existence of compatible tower and the spectrality of \(\mu _{\{M_n\},\{B_n\}}\) are equivalent.

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Acknowledgements

The authors would like to thank the referees for their valuable suggestions, and also Professor Li-**ang An, Guo-Tai Deng for many helpful discussions on the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, People’s Republic of China

    Zi-Chao Chi & Jian-Feng Lu

  2. Department of Applied Mathematics, Anhui University of Technology, Ma’anshan, 243002, People’s Republic of China

    Min-Min Zhang

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  1. Zi-Chao Chi
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  2. Jian-Feng Lu
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  3. Min-Min Zhang
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Correspondence to Min-Min Zhang.

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This work was supported by the National Natural Science Foundation of China 11971194.

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Chi, ZC., Lu, JF. & Zhang, MM. A Class of Spectral Moran Measures Generated by the Compatible Tower. J Geom Anal 34, 201 (2024). https://doi.org/10.1007/s12220-024-01646-1

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