# Spectral property of the planar self-affine measures with three-element digit sets

Spectral property of the planar self-affine measures with three-element digit sets AbstractLet the self-affine measure μM,D{\mu_{M,D}}be generated by an expanding real matrix M=diag⁡(ρ1-1,ρ2-1){M=\operatorname{diag}(\rho_{1}^{-1},\rho_{2}^{-1})}and an integer digit setD={(0,0)t,(α1,α2)t,(β1,β2)t}{D=\{(0,0)^{t},(\alpha_{1},\alpha_{2})^{t},(\beta_{1},\beta_{2})^{t}\}}with α1⁢β2-α2⁢β1≠0{\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0}. In this paper, the sufficient and necessary conditions for L2⁢(μM,D){L^{2}(\mu_{M,D})}to contain an infinite orthogonal set of exponential functions are given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

# Spectral property of the planar self-affine measures with three-element digit sets

Forum Mathematicum, Volume 32 (3): 9 – May 1, 2020
9 pages

/lp/de-gruyter/spectral-property-of-the-planar-self-affine-measures-with-three-8iSJqawkfR
Publisher
de Gruyter
© 2020 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2019-0223
Publisher site
See Article on Publisher Site

### Abstract

AbstractLet the self-affine measure μM,D{\mu_{M,D}}be generated by an expanding real matrix M=diag⁡(ρ1-1,ρ2-1){M=\operatorname{diag}(\rho_{1}^{-1},\rho_{2}^{-1})}and an integer digit setD={(0,0)t,(α1,α2)t,(β1,β2)t}{D=\{(0,0)^{t},(\alpha_{1},\alpha_{2})^{t},(\beta_{1},\beta_{2})^{t}\}}with α1⁢β2-α2⁢β1≠0{\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0}. In this paper, the sufficient and necessary conditions for L2⁢(μM,D){L^{2}(\mu_{M,D})}to contain an infinite orthogonal set of exponential functions are given.

### Journal

Forum Mathematicumde Gruyter

Published: May 1, 2020

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