Spectrality of self-affine Sierpinski-type measures on R2

By Xin-Rong Dai, Xiaoye Fu, Zhiwei Yan. 2020

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Abstract In this paper, we study the spectral property of a class of self-affine measures μ R , D on R 2 generated by the iterated function system { ϕ d ( ⋅ ) = R − 1 ( ⋅ + d ) } d ∈ D associated with the real expanding matrix R = ( b 1 0 0 b 2 ) and the digit set D = { ( 0 0 ) , ( 1 0 ) , ( 0 1 ) } . We show that μ R , D is a spectral measure if and only if 3 | b i , i = 1 , 2 . This extends the result of Deng and Lau [J. Funct. Anal., 2015], where they considered the case b 1 = b 2 . And we also give a tree structure for any spectrum of μ R , D by providing a decomposition property on it.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a6bd574bc22ae74f84930a138878e9bf60bd5bf9