On the spectra of a class of Moran measures

Ming-Liang Chen; Jian Cao; Jia-Lin Wang; Ye Wang

doi:10.1515/forum-2023-0029

Published by De Gruyter June 1, 2023

On the spectra of a class of Moran measures

Ming-Liang Chen , Jian Cao , Jia-Lin Wang and Ye Wang

From the journal Forum Mathematicum

https://doi.org/10.1515/forum-2023-0029

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Abstract

Let { A n } n = 1 ∞ be a sequence of expanding matrices with A n ∈ M 2 ⁢ ( ℤ ) , and let { D n } n = 1 ∞ be a sequence of three-element digit sets with { x ∈ ( 0 , 1 ) 2 : ∑ d ∈ D n e 2 ⁢ π ⁢ i ⁢ 〈 d , x 〉 = 0 } = { ± 1 3 ⁢ ( 1 , i ) t } , i ∈ { 1 , 2 } . The associated Moran measure generated by the infinite convolution

μ { A n } , { D n } = δ A 1 - 1 ⁢ D 1 * δ A 1 - 1 ⁢ A 2 - 1 ⁢ D 2 * δ A 1 - 1 ⁢ A 2 - 1 ⁢ A 3 - 1 ⁢ D 3 * ⋯ .

In this paper, we give some necessary and sufficient conditions for μ { A n } , { D n } to be a spectral measure under some suitable conditions on A n and D n .

Keywords: Moran measure; spectral measure; spectrum; Fourier transform

MSC 2020: 28A25; 28A80; 42C05; 46C05

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12071125

Award Identifier / Grant number: 11831007

Award Identifier / Grant number: 11971500

Award Identifier / Grant number: 12061010

Funding statement: The research is supported in part by the NNSF of China (Nos. 12071125, 11831007, 11971500 and 12061010), the Science and Technology Research Project of Jiangxi Provincial Department of Education (No. GJJ2201244), the Doctoral Scientific Research Foundation of Gannan Normal University (No. BSJJ202241).

Acknowledgements

The authors would like to thank the referee for his/her many valuable comments and suggestions.

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Received: 2023-02-04

Revised: 2023-03-17

Published Online: 2023-06-01

Published in Print: 2024-03-01

On the spectra of a class of Moran measures

Abstract

Acknowledgements

References

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