Home Mathematics Spectral properties of certain Moran measures with consecutive and collinear digit sets
Article
Licensed
Unlicensed Requires Authentication

Spectral properties of certain Moran measures with consecutive and collinear digit sets

  • Hai-Hua Wu , Yu-Min Li EMAIL logo and Xin-Han Dong
Published/Copyright: February 5, 2020

Abstract

Let the 2×2 expanding matrix Rk be an integer Jordan matrix, i.e., Rk=diag(rk,sk) or Rk=J(pk), and let Dk={0,1,,qk-1}v with v=(1,1)T and 2qkpk,rk,sk for each natural number k. We show that the sequence of Hadamard triples {(Rk,Dk,Ck)} admits a spectrum of the associated Moran measure provided that lim infk2qkRk-1<1.

MSC 2010: 28A80; 42C05

Communicated by Siegfried Echterhoff


Award Identifier / Grant number: 11571099

Award Identifier / Grant number: 11701166

Award Identifier / Grant number: 11831007

Funding statement: The research is supported in part by the NNSF of China (No. 11701166, 11831007, 11571099).

Acknowledgements

The authors are grateful to the referees for going through the paper in great detail and for providing many valuable suggestions to improve the presentation.

References

[1] L.-X. An, X.-Y. Fu and C.-K. Lai, On spectral Cantor–Moran measures and a variant of Bourgain’s sum of sine problem, Adv. Math. 349 (2019), 84–124. 10.1016/j.aim.2019.04.014Search in Google Scholar

[2] L.-X. An, L. He and X.-G. He, Spectrality and non-spectrality of the Riesz product measures with three elements in digit sets, J. Funct. Anal. 277 (2019), no. 1, 255–278. 10.1016/j.jfa.2018.10.017Search in Google Scholar

[3] L.-X. An and X.-G. He, A class of spectral Moran measures, J. Funct. Anal. 266 (2014), no. 1, 343–354. 10.1016/j.jfa.2013.08.031Search in Google Scholar

[4] L.-X. An, X.-G. He and K.-S. Lau, Spectrality of a class of infinite convolutions, Adv. Math. 283 (2015), 362–376. 10.1016/j.aim.2015.07.021Search in Google Scholar

[5] L.-X. An, X.-G. He and H.-X. Li, Spectrality of infinite Bernoulli convolutions, J. Funct. Anal. 269 (2015), no. 5, 1571–1590. 10.1016/j.jfa.2015.05.008Search in Google Scholar

[6] X.-R. Dai, When does a Bernoulli convolution admit a spectrum?, Adv. Math. 231 (2012), no. 3–4, 1681–1693. 10.1016/j.aim.2012.06.026Search in Google Scholar

[7] X.-R. Dai, Spectra of Cantor measures, Math. Ann. 366 (2016), no. 3–4, 1621–1647. 10.1007/s00208-016-1374-5Search in Google Scholar

[8] X.-R. Dai, X.-G. He and C.-K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013), 187–208. 10.1016/j.aim.2013.04.016Search in Google Scholar

[9] X.-R. Dai, X.-G. He and K.-S. Lau, On spectral N-Bernoulli measures, Adv. Math. 259 (2014), 511–531. 10.1016/j.aim.2014.03.026Search in Google Scholar

[10] X.-R. Dai and Q.-Y. Sun, Spectral measures with arbitrary Hausdorff dimensions, J. Funct. Anal. 268 (2015), no. 8, 2464–2477. 10.1016/j.jfa.2015.01.005Search in Google Scholar

[11] Q.-R. Deng and K.-S. Lau, Sierpinski-type spectral self-similar measures, J. Funct. Anal. 269 (2015), no. 5, 1310–1326. 10.1016/j.jfa.2015.06.013Search in Google Scholar

[12] D. Dutkay, J. Haussermann and C.-K. Lai, Hadamard triples generate self-affine spectral measures, Trans. Amer. Math. Soc. 371 (2019), no. 2, 1439–1481. 10.1090/tran/7325Search in Google Scholar

[13] D. Dutkay and P. Jorgensen, Analysis of orthogonality and of orbits in affine iterated function systems, Math. Z. 256 (2007), no. 4, 801–823. 10.1007/s00209-007-0104-9Search in Google Scholar

[14] D. Dutkay and C.-K. Lai, Uniformity of measures with Fourier frames, Adv. Math. 252 (2014), 684–707. 10.1016/j.aim.2013.11.012Search in Google Scholar

[15] K. Falconer, Fractal Geometry, John Wiley, Chichester, 1990. Search in Google Scholar

[16] X.-Y. Fu, X.-G. He and K.-S. Lau, Spectrality of self-similar tiles, Constr. Approx. 42 (2015), no. 3, 519–541. 10.1007/s00365-015-9306-2Search in Google Scholar

[17] Y.-S. Fu, X.-G. He and Z.-X. Wen, Spectra of Bernoulli convolutions and random convolutions, J. Math. Pures Appl. (9) 116 (2018), 105–131. 10.1016/j.matpur.2018.06.002Search in Google Scholar

[18] B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Analysis 16 (1974), 101–121. 10.1016/0022-1236(74)90072-XSearch in Google Scholar

[19] L. He and X.-G. He, On the Fourier orthonormal bases of Cantor–Moran measures, J. Funct. Anal. 272 (2017), no. 5, 1980–2004. 10.1016/j.jfa.2016.09.021Search in Google Scholar

[20] X.-G. He, C.-K. Lai and K.-S. Lau, Exponential spectra in L2(μ), Appl. Comput. Harmon. Anal. 34 (2013), no. 3, 327–338. 10.1016/j.acha.2012.05.003Search in Google Scholar

[21] T.-Y. Hu and K.-S. Lau, Spectral property of the Bernoulli convolutions, Adv. Math. 219 (2008), no. 2, 554–567. 10.1016/j.aim.2008.05.004Search in Google Scholar

[22] J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. 10.1512/iumj.1981.30.30055Search in Google Scholar

[23] J. Jacod and P. Protter, Probability Essentials, 2nd ed., Universitext, Springer, Berlin, 2003. 10.1007/978-3-642-55682-1Search in Google Scholar

[24] P. Jorgensen, S. Pedersen and F. Tian, Spectral theory of multiple intervals, Trans. Amer. Math. Soc. 367 (2015), no. 3, 1671–1735. 10.1090/S0002-9947-2014-06296-XSearch in Google Scholar

[25] P. Jorgensen and S. Pedersen, Dense analytic subspaces in fractal L2-spaces, J. Anal. Math. 75 (1998), 185–228. 10.1007/BF02788699Search in Google Scholar

[26] M. Kolountzakis and M. Matolcsi, Tiles with no spectra, Forum Math. 18 (2006), no. 3, 519–528. 10.1515/FORUM.2006.026Search in Google Scholar

[27] I. Łaba and Y. Wang, On spectral Cantor measures, J. Funct. Anal. 193 (2002), no. 2, 409–420. 10.1006/jfan.2001.3941Search in Google Scholar

[28] C.-K. Lai, K.-S. Lau and H. Rao, Spectral structure of digit sets of self-similar tiles on 1, Trans. Amer. Math. Soc. 365 (2013), no. 7, 3831–3850. 10.1090/S0002-9947-2013-05787-XSearch in Google Scholar

[29] J.-L. Li, On the μM,D-orthogonal exponentials, Nonlinear Anal. 73 (2010), no. 4, 940–951. 10.1016/j.na.2010.04.017Search in Google Scholar

[30] J.-L. Li, Spectra of a class of self-affine measures, J. Funct. Anal. 260 (2011), no. 4, 1086–1095. 10.1016/j.jfa.2010.12.001Search in Google Scholar

[31] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Stud. Adv. Math. 44, Cambridge University, Cambridge, 1995. 10.1017/CBO9780511623813Search in Google Scholar

[32] R. Strichartz, Mock Fourier series and transforms associated with certain Cantor measures, J. Anal. Math. 81 (2000), 209–238. 10.1007/BF02788990Search in Google Scholar

[33] R. Strichartz, Convergence of mock Fourier series, J. Anal. Math. 99 (2006), 333–353. 10.1007/BF02789451Search in Google Scholar

[34] M.-W. Tang and F.-L. Yin, Spectrality of Moran measures with four-element digit sets, J. Math. Anal. Appl. 461 (2018), no. 1, 354–363. 10.1016/j.jmaa.2018.01.018Search in Google Scholar

[35] T. Tao, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), no. 2–3, 251–258. 10.4310/MRL.2004.v11.n2.a8Search in Google Scholar

[36] Z.-Y. Wang, X.-H. Dong and Z.-S. Liu, Spectrality of certain Moran measures with three-element digit sets, J. Math. Anal. Appl. 459 (2018), no. 2, 743–752. 10.1016/j.jmaa.2017.11.006Search in Google Scholar

Received: 2019-09-05
Revised: 2020-01-05
Published Online: 2020-02-05
Published in Print: 2020-05-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 4.2.2026 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2019-0248/pdf
Scroll to top button